The issue of establishing a child's paternal relationship goes beyond who to address the Father's Day card to.
In Halacha, establishing a child's father affects who the child may or may not marry since if the child is indeed a child of the father in question, the father's other children would be prohibited for him to marry because of arayos, the cardinal sin of forbidden marital relationships.
However, as we will see, in order to determine a child's father in the eyes of Halacha, there are certain qualifications. The Gemara says in Yevamos:
אָמַר רָבָא, הָא דַּאֲמוּר רַבָּנַן: אֵין אָב לְגוֹי...שְׁמַע מִינַּהּ: אַפְקוֹרֵי אַפְקְרֵיהּ רַחֲמָנָא לְזַרְעֵיהּ, דִּכְתִיב: ״בְּשַׂר חֲמוֹרִים בְּשָׂרָם וְזִרְמַת סוּסִים זִרְמָתָם״...
תָּא שְׁמַע: גֵּר שֶׁהָיָה לֵידָתוֹ בִּקְדוּשָּׁה וְהוֹרָתוֹ שֶׁלֹּא בִּקְדוּשָּׁה — יֵשׁ לוֹ שְׁאֵר הָאֵם, וְאֵין לוֹ שְׁאֵר הָאָב. כֵּיצַד? נָשָׂא אֲחוֹתוֹ מִן הָאֵם — יוֹצִיא. מִן הָאָב — יְקַיֵּים.
Rava said: With regard to that which the Sages said, that a gentile has no patrilineage... Learn from this that the Merciful One dispossesses the male gentile of his offspring, as it is written with regard to Egyptians: “Whose flesh is the flesh of donkeys, and whose semen is the semen of horses” (Ezekiel 23:20), i.e., the offspring of a male gentile is considered no more related to him than the offspring of donkeys and horses...
Come and hear another proof: A convert whose birth was in sanctity but whose conception was not in sanctity has maternal kinship, i.e., his relationship to his mother’s relatives is recognized.
However, he does not have paternal kinship. How so? If he married his maternal half sister, who was born before him and converted, he must divorce her. Although by Torah law they are considered unrelated, the Sages rendered it prohibited for them to marry, lest he marry a maternal half sister who was born after him and is forbidden to him. If she is his paternal half sister, he may maintain her as his wife. If he married his father’s maternal half sister, he must divorce her.
הא דאמור רבנן אין אב למצרי - ואפילו היכא דליכא למימר כקטן שנולד דמי כגון הורתו שלא בקדושה ולידתו בקדושה דיש לו שאר מן האם כשאר ישראל גמור ואין לו שאר מן האב: לא תימא משום דשטופי זימה הוא ולא ידעינן - אבוה מנו אבל היכא דידיע כגון היו אביו ואמו חבושים בבית האסורים דלא נבעלה לאחר חיישינן ואי נולד בקדושה דליכא למימר כקטן שנולד ליהוי ליה נמי שאר אב דהא ודאי אבוה הוא:
A gentile has no patrilineage - Even in a case where the principle of the conversion deeming the child like a "newborn" would not apply, such as if the child was conceived while the father was a gentile but subsequently born as a Jew (when the mother converts during the preganancy), the father loses his connection to the son [because of the concept that the Merciful One dispossesses his seed.]
Rav Moshe Shmuel Shapiro deduces from Rashi the framework of how the paternal relationship is created and subsequently severed in the scenario that there is a conversion.
Once a child is born to the father, the relationship is firmly established. Rav Moshe Shmuel connects the birth of the child to connect to the child by dint of his active role of through the conception of the child months before.
If the child would subsequently to convert to Judasim, Chazal say that the conversion makes him "like a newborn child", thereby reversing the conception-birth connection attributed to the father. This "rebirth" is mentioned in the nomenclature of Chazal earlier in Maseches Yevamos.
וְגֵר שֶׁנִּתְגַּיֵּיר כְּקָטָן שֶׁנּוֹלַד דָּמֵי
And the legal status of a convert who just converted is like that of a child just born, and all previous family ties become irrelevant.
But in a case where the child is conceived when the father is a gentile (thus establishing the aforementioned paternal connection) and during pregnancy the mother (and by extension the child) converts, the principle of "becoming like a newborn child" would not apply as the child is still being formed, it would logically follow that the connection to the father should remain in place.
To this point Rashi is telling us, that the relationship to the father becomes severed because of this other Halachic principle that "the Torah dispossesses the seed of the father," which says that although the relationship created by dint of his active role of through the conception of the child months before, there is another consideration - that of the father's relationship to the child throughout the formation of the child. Since the mother converted, the Halachic principle of "the Torah dispossesses his seed" applies and severs his relationship to the child.
It emerges, says Rav Moshe Shmuel Shapiro, that there are two parts in creating a paternal connection to a child (according to my understanding)
1. The father's active role of bringing about the birth of the child months before by conception. Immediately after the child is born, the child becomes his son, and he, the father.
2. Throughout the pregnancy, each passing day of the formation of the child, the paternal relationship is gradually created.
The clear conclusion of the Gemara is that in order for the laws of arayos to apply to paternal relatives, both of these relationships have to remain intact.
Rav Moshe Shmuel Shapiro posits that these two connections (yichusin) are alluded to in Onkelos in Parshas Be'Haaloshecha:
(יב) הֶאָנֹכִ֣י הָרִ֗יתִי אֵ֚ת כׇּל־הָעָ֣ם הַזֶּ֔ה אִם־אָנֹכִ֖י יְלִדְתִּ֑יהוּ כִּֽי־תֹאמַ֨ר אֵלַ֜י שָׂאֵ֣הוּ בְחֵיקֶ֗ךָ כַּאֲשֶׁ֨ר יִשָּׂ֤א הָאֹמֵן֙ אֶת־הַיֹּנֵ֔ק עַ֚ל הָֽאֲדָמָ֔ה אֲשֶׁ֥ר נִשְׁבַּ֖עְתָּ לַאֲבֹתָֽיו׃
(12) Did I produce all these people, did I give birth to them, that You should say to me, ‘Carry them in your bosom as a caregiver carries an infant,’ to the land that You have promised on oath to their fathers?
Onkelos writes:
(יב) הֲאַב אֲנָא לְכָל עַמָּא הָדֵין אִם בְּנַי אִנּוּן דְּאָמְרֵת לִי סוֹבַרְהִי בְּתָקְפָּךְ כְּמָא דִמְסוֹבַר תֻּרְבְּיָנָא יָת יַנְקָא עַל אַרְעָא דִּי קַיֶּמְתָּא לַאֲבָהָתוֹהִי:
(12) Am I a father to this entire people? Are they my children that You say to me, carry it [with might] in your lap as a nurse carries an infant, to the land which You swore to their forefathers?
The first part of the passuk refers to the paternal relationship created throughout pregnancy (the word "harisi" relates to the word "hirayon", which means "pregnancy" and particularly the idea of initiating the preganancy as the Ramban here writes:
(1) DID I ‘HARITHI’ ALL THIS PEOPLE? In the opinion of Onkelos who rendered it: “Am I ‘the father’ of all this people?”... a father is called "horeh" because it is he who brings about the pregnancy ("hirayon") [of the mother].
The second part of the passuk is referring to the father's relationship created at the time of the birth when the child is considered his son, and by extension, he the father.
Until this point is just an elucidated explanation of Rav Moshe Shmuel Shapiro's understanding of Onkelos.
I'd like to add some points and questions on his vort.
1. In detailing these two paternal aspects, Moshe Rabbeinu isn't simply just using an abstract "lomdishe'" idea to make his point. He is saying that in order to expect a person to have a strong level of care and concern for someone or something, there needs to be a connection between them.
Inasmuch as that connection is established is the level of responsibility that can be expected.
Moshe Rabbeinu is telling Hashem that what is being asked of him in relation to his connection to Klal Yisrael is disproportionate; he has neither the paternal connection that grows throughout pregnancy, nor the connection that is created upon birth of the child from his contribution of the conceiver, and is thus asking Hashem to absolve him of being responsible in finding a solution for their plight.
See Or HaChaim who says that although Moshe was considered a "father" in some spiritual respects, it did not translate into the nature of responsibility he was asked to take for them. He writes:
(א) האנכי הריתי וגו'. דבר בסדר זה, יתבאר על פי דבריהם ז"ל (סנהדרין יט:) שכל המלמד בן חבירו תורה כאלו ילדו, גם אמרו ז"ל (תיקונים ע') שכל נשמות עם בני ישראל דור המדבר היו ענפי נשמתו של משה רבינו והוא היה להם לאב, לזה לא אמר האב אני לעם הזה או אם בני הם, כי כן הוא כפי האמת שהוא היה אב לנשמותם ונקראים בניו גם כן לצד שלמדם תורה, אלא האנכי הריתי פירוש כנגד צד היותם ענפיו אמר מה בכך כיון שלא הורה גבר מהם לא יתחייב בטיפולם, וכנגד שנקראים בניו לצד שלמדם תורה אמר אם אנכי ילדתיהו כי דוקא בן הבא דרך לידה לא הבא מאמצעות לימוד תורה:
(1) האנכי הריתי את כל העם הזה, "Did I conceive this whole people?" The reason that Moses made such an apparently inappropriate comparison is based on Sanhedrin 19 where we are told that whoever teaches the son of his friend Torah is considered as if he had given birth to him. There is also a comment in Tikkuney Hazohar 70 that all the souls of the people who marched through the desert were "branches" of Moses' own soul and that he was considered the father of all those souls.
Concerning the relationship between Moshe and Klal Yisrael as them being "branches", Moses asked האנכי הריתי, "did I conceive them?", what does this point matter, only one who actually concieved them should be held responsible to take care of them.
Concerning the relationship between Moshe and Klal Yisrael as them being his "children" from teaching them Torah, he asked אם אנכי ילדתיהו "did I give birth to them?", it is only an actual father who should be held responsible for their actions, not a "father" who is brought about by means of teaching them Torah.
When Hashem does urge Moshe to nevertheless take responsibility, it is in response to his other complaint that the burden was too heavy, not this one.
(טז) וַיֹּ֨אמֶר יהוה אֶל־מֹשֶׁ֗ה אֶסְפָה־לִּ֞י שִׁבְעִ֣ים אִישׁ֮ מִזִּקְנֵ֣י יִשְׂרָאֵל֒ אֲשֶׁ֣ר יָדַ֔עְתָּ כִּי־הֵ֛ם זִקְנֵ֥י הָעָ֖ם וְשֹׁטְרָ֑יו וְלָקַחְתָּ֤ אֹתָם֙ אֶל־אֹ֣הֶל מוֹעֵ֔ד וְהִֽתְיַצְּב֥וּ שָׁ֖ם עִמָּֽךְ׃
(16) Then יהוה said to Moses, “Gather for Me seventy of Israel’s elders of whom you have experience as elders and officers of the people, and bring them to the Tent of Meeting and let them take their place there with you.
(א) אספה לי. הֲרֵי תְשׁוּבָה לִתְלוּנָתְךָ שֶׁאָמַרְתָּ לֹא אוּכַל אָנֹכִי לְבַדִּי;
(1) אספה לי GATHER UNTO ME [SEVENTY MEN OF THE ELDERS OF ISRAEL] — "This" — the Lord said — “is the reply to your complaint which you uttered (v. 14) I am not able to bear all this people alone”’ (cf. Sifrei Bamidbar 92)
(However, according to the alternate explanation found in the Or HaChaim, it can be argued that the original complaint was also being addressed, see there)
2. My friend, Rav Yitzchak Iyzik Tendler, asked on the point of Rav Moshe Shmuel Shapiro that there is a paternal relationship created during pregnancy, that it seems to be at odds with a Gemara in Megilla.
״כִּי אֵין לָהּ אָב וָאֵם״ — ״וּבְמוֹת אָבִיהָ וְאִמָּהּ״ לְמָה לִי? אָמַר רַב אַחָא: עִיבְּרַתָּה — מֵת אָבִיהָ, יְלָדַתָּה — מֵתָה אִמָּהּ.
“And when her father and mother were dead, Mordecai took her for his own daughter”? Rav Aḥa said: This repetition indicates that when her mother became pregnant with her, her father died, and when she gave birth to her, her mother died, so that she did not have a mother or a father for even a single day.
We see from here that although the father was not alive at the birth of the child, he was (presumably) alive for a period of time during the pregnancy, and according to Rav Moshe Shmuel Shapiro, there was a point in time when she had a "father" relationship, if so why does the passuk say that she didn't have any father relationship?
The Maharal, in his commentary on Esther, touches on this point. Maharal writes:
"כי אין לה אב ואם", וכתיב "במות אביה ואמה" למה לי. אמר רב חסדא, עוברתה אמה, מת אביה. ילדתה אמה, מתה אמה. ונראה שדרשו כך, מפני כי "ובמות אביה ואמה" הוא יתור, ולא הוי צריך לכתוב, לכך דרש "במות אביה" היינו בשעה שנעשה לה אב, והיינו "עברתה מת אביה", שבאותו שעה נעשה אביה. ובשעה שנעשית לה אמה, וזה בשעה שנולדה. דבשעת העיבור, כיון שלא פירש הולד (-ממנו-) [ממנה] לא נקרא עדיין "אמה", מה שאין כן אצל האב, שמיד שנתעברה ממנו נקרא "אביה". ואצל האם קיימא לן "עובר ירך אמו" (גיטין כג:), וכשילדתה מתה אמה.
I don't fully understand the Maharal, because if the father relationship is reckoned at the time of pregnancy, why does the passuk say "for she had no father?"
Perhaps we can answer this question based on a nuance made by the Ralbag in the understand the drasha of Chazal we can answer this question. Ralbag (brought by Manos HaLevi) writes:
(א) ויהי אמן את הדסה היא אסתר בת דדו כי אין לה אב ואם והנערה יפת תאר וטובת מראה ובמות אביה ואמה לקחה מרדכי לו לבת:
מצאתי אל הה"ר יהודה ן' שוש"ן ז"ל שהאריך בזה וז"ל ויש לי לדקדק כפי קבלתם ז"ל שאמרו עברתה אמה מת אביה וכו' כי אמר הכתוב כי מרדכי גדלתה מימי ינקותה וז"ט ויהי אומן כי לשון זה לא מצאתיו רק על ימי הגדול והיניקה (במדבר י"א י"ב) כאשר ישא. האומן את היונק והיו מלכים אומניך ותהי לו לאומנת וג' עדים כמאה.
והרלב"ג הרגיש בזה שפי' ויהי כאלו אמר והיה אומן כי לפי פירושנו זה לא היה עתה אומן אותה כי כבר היתה מגודלת ובלשון הכתוב ויהי אומן בימים שעברו כי אין לה זמן בעולם שהיה לה אב ואם, אמנם היה לה עת ורגע שהיה לה אם ולא אב והוא בעת לידתה ולא יותר שאלו היתה אמה אחריה מה טעם ויהי אומן, והנה אמה בצדה ע"כ.
Ralbag says that the passuk is saying that there was never a time when Esther had both parents at the same time. Ralbag only deals with the mother, that there was a period of time when she had a mother (at the time of birth) but it was without the father.
Perhaps we can say that the Gemara can also have the opposite implication as well. When her mother was pregnant, there was a period of time when she had a father, since an aspect of fatherhood does start at pregnancy, but no mother (since as Ralbag says, the mother-connection is not considered during the pregnancy, rather at the birth)
Another mareh makom about this debate as to whether "fatherhood" starts at the time of pregnancy or birth in regards to inheritance laws, like determining who the bechor is, is found in the Gra to Parshas Ki Teitzei. See Peninim MiShulchan Govoha where there are two opposite accounts of what the Gra said, whether it is determined by which child is conceived/pregnancy began first or who was actually born first.
(In the footnote to the shtikl there, there is a tzad that when there are twins who are born from a single drop that splits, then even according to the opinion that we reckon the time of the conception/pregnancy, since in that regard there is a "tie" we would then revert to go after the actual birth. (This would seem to be somewhat of a smach to R' Moshe Shmuel Shapiro that there are two considerations in making the fatherhood, otherwise one would be arbitrary.)