There is an undying misconception spread by many people that the Sages and Oral Torah do not have a "gender binary", many take this further and claim that our patriarchs and matriarchs are not men and women. These claims are based in general misunderstanding of the words of Our Sages and of the Torah in general.
True definition of Saris, Aylonit, Androgynos, and Tumtum.
The terms "saris", "aylonit", "androgynos", and "tumtum" refer to physical characteristics, a saris being an infertile male, an aylonit being an infertile female, an androgynos referring to a person with both sexual characteristics, and a tumtum referring to a person with indeterminate or underdeveloped sexual characteristics. They are said by some to be different genders, but they are not.
(ב) כֵּיצַד שָׁוֶה לַאֲנָשִׁים: מְטַמֵּא בְּלֹבֶן כַּאֲנָשִׁים, וְזוֹקֵק לְיִבּוּם כַּאֲנָשִׁים, וּמִתְעַטֵּף וּמִסְתַּפֵּר כַּאֲנָשִׁים, וְנוֹשֵׂא אֲבָל לֹא נִשָּׂא כַּאֲנָשִׁים, וְחַיָּב בְּכָל מִצְוֹת הָאֲמוּרוֹת בַּתּוֹרָה כַּאֲנָשִׁים:
In what ways is s/he similar to men? Like a man, s/he is considered unclean through semen; is required to perform yibbum (levirate marriage) like a man; dresses and cuts hair like a man; marries others and is not married off, like a man; and is obliged to perform all the commandments in the Torah, like a man
(ו) סְרִיס חַמָּה כֹּהֵן שֶׁנָּשָׂא בַת יִשְׂרָאֵל, מַאֲכִילָהּ בַּתְּרוּמָה. רַבִּי יוֹסֵי וְרַבִּי שִׁמְעוֹן אוֹמְרִים, אַנְדְּרוֹגִינוֹס כֹּהֵן שֶׁנָּשָׂא בַת יִשְׂרָאֵל, מַאֲכִילָהּ בַּתְּרוּמָה. רַבִּי יְהוּדָה אוֹמֵר, טֻמְטוּם שֶׁנִּקְרַע וְנִמְצָא זָכָר, לֹא יַחֲלֹץ, מִפְּנֵי שֶׁהוּא כְסָרִיס. אַנְדְּרוֹגִינוֹס נוֹשֵׂא, אֲבָל לֹא נִשָּׂא. רַבִּי אֱלִיעֶזֶר אוֹמֵר, אַנְדְּרוֹגִינוֹס חַיָּבִים עָלָיו סְקִילָה, כְּזָכָר:
(6) If a priest who was eunuch by nature married the daughter of an Israelite, he confers upon her the right to eat terumah. Rabbi Yose and Rabbi Shimon stated: if a priest who was an hermaphrodite married the daughter of an Israelite, he confers upon her the right to eat terumah. Rabbi Judah stated: if a tumtum was opened up and found to be a male, he may not perform halitzah, because he has the same status as a eunuch. The hermaphrodite may marry [a wife] but may not be married [by a man]. Rabbi Eliezer stated: concerning the hermaphrodite, [the one who has relations with him] is liable to be stoned like one [who has relations with] a male.
The Sages clearly saw an androgynos as to be required to follow most halakha generally for males, and attempts to explain them in this catagory by establishing the halakha for them based on their similarities and differences to males and females, and generally treated them as more masculine than feminine.
Aylonit and Saris are said to represent other genders, besides male and female, which is a misrepresentation of the meaning of both. Saris and Aylonit are catagories of men and women who are underdeveloped.
(ח) הַמּוֹצִיא אֶת אִשְׁתּוֹ מִשּׁוּם אַיְלוֹנִית, רַבִּי יְהוּדָה אוֹמֵר, לֹא יַחֲזִיר. וַחֲכָמִים אוֹמְרִים, יַחֲזִיר. נִשֵּׂאת לְאַחֵר וְהָיוּ לָהּ בָּנִים הֵימֶנּוּ, וְהִיא תוֹבַעַת כְּתֻבָּתָהּ, אָמַר רַבִּי יְהוּדָה, אוֹמְרִים לָהּ, שְׁתִיקוּתִיךְ יָפָה לִיךְ מִדִּבּוּרִיךְ:
(8) A man divorces his wife because she is an aylonit: Rabbi Judah says he may not remarry her, But the sages say that he may remarry her. She marries someone else and has children from him and then demands her ketubah settlement [from her first husband]: Rabbi Judah said, they say to her, “Your silence is better than your speaking.”
Men who divorce infertile women are allowed to remarry their wives, they are explicitly seen as women, not a different catagory.
(ט) בַּת עֶשְׂרִים שָׁנָה שֶׁלֹּא הֵבִיאָה שְׁתֵּי שְׂעָרוֹת, תָּבִיא רְאָיָה שֶׁהִיא בַת עֶשְׂרִים שָׁנָה, וְהִיא אַיְלוֹנִית, לֹא חוֹלֶצֶת וְלֹא מִתְיַבֶּמֶת. בֶּן עֶשְׂרִים שָׁנָה שֶׁלֹּא הֵבִיא שְׁתֵּי שְׂעָרוֹת, יָבִיא רְאָיָה שֶׁהוּא בֶן עֶשְׂרִים שָׁנָה, וְהוּא סָרִיס, לֹא חוֹלֵץ וְלֹא מְיַבֵּם, אֵלּוּ דִּבְרֵי בֵית הִלֵּל. בֵּית שַׁמַּאי אוֹמְרִים, זֶה וָזֶה בְּנֵי שְׁמֹנֶה עֶשְׂרֵה. רַבִּי אֱלִיעֶזֶר אוֹמֵר, הַזָּכָר כְּדִבְרֵי בֵית הִלֵּל, וְהַנְּקֵבָה כְּדִבְרֵי בֵית שַׁמַּאי, שֶׁהָאִשָּׁה מְמַהֶרֶת לָבֹא לִפְנֵי הָאִישׁ:
(9) If a woman at the age of twenty did not bring forth two hairs, she must bring evidence that she is twenty years of age and she is an aylonit, she doesn't perform halitzah or yibbum. If a man at the age of twenty years did not produce two hairs, he must bring evidence that he is twenty years old and he becomes confirmed as a saris and he doesn't perform halitzah or yibbum, the words of Bet Hillel. Bet Shammai says: with both of them at the age of eighteen. Rabbi Eliezer says: In the case of the male, according to the words of Bet Hillel, while in that of the female, in accordance with the words of Bet Shammai, since a woman matures earlier than a man.
The definition of aylonit and saris itself says that it is a woman and a man respectively who do not possess proper sexual characteristics for their age. They are not separate from women and men.
רבי אמי אברהם ושרה טומטמין היו שנאמר (ישעיהו נא, א) הביטו אל צור חוצבתם ואל מקבת בור נוקרתם וכתיב (ישעיהו נא, ב) הביטו אל אברהם אביכם ואל שרה תחוללכם
Rabbi Ami said: Abraham and Sarah were originally tumtumin, people whose sexual organs are concealed and not functional, as it is stated: “Look to the rock from where you were hewn, and to the hole of the pit from where you were dug” (Isaiah 51:1), and it is written in the next verse: “Look to Abraham your father and to Sarah who bore you” (Isaiah 51:2), which indicates that sexual organs were fashioned for them, signified by the words hewn and dug, over the course of time.
Tumtum comes from the Hebrew word for "hidden", as in their sexual characteristics are hidden. It is also not a different catagory.
Incorrect conclusions
This is used to claim that Sarah Imeinu and Avraham Avinu are a different gender, not man and woman, however as explained above these terms are not separate genders. This also implies that Tumtum and Aylonit are not mutually exclusive, as Sarah is called both, which if they were genders would not be true, but if they were descriptors they would be.
The word used here is na'arah, however it does not have the extra hei to signify the vowel. The extra hei is not grammatically required, so it can be read as na'ar (masculine) in the consonantal text, however the vowels used to understand the Torah show that this word is actually na'arah.
The same occurs here, however the verb is even conjugated feminine, showing is is in fact na'arah as opposed to na'ar. In most cases in the Torah the Hei is omitted, so this is not strange, but consistent with most of the rest of the Torah.
In an attempt to prove that Yakov Avinu is not a man, these verses are cited showing his nature that is not the Western stereotypical definition of "masculine". Not only is this ironic, as the people who claim to be against Western gender roles are very happy to use them to prove a point, but also is Eurocentric, focusing on modern European interpretations of masculinity, and generally silly, as not being as "masculine" as another man does not prove one is not a man.
In an attempt to prove Yakov Avinu did "drag", Koatz cite Yakov dressing as Esav. This is a case of reading meaning into something where it is not there. It is simply a man dressing as another man as a disguise.
The terms used to describe Yosef Avinu and Rachel Imeinu in these verses are the same. This is read to mean that they are both queer. However the Hebrew terms here are very androgynous, and the English reading is simply for poetic effect.
The coat is referred to as "flamingly gay, femme and sparkly" by Koatz, however this uses the English translation of a vague Hebrew term, where Rabbis define it as ranging in meaning from embroidered (Ibn Ezra), colorful (JPS, Radak), striped (Rabbeinu Bahya), a robe (Rashbam, Targum Yonatan, Targum Yerushalayim), or linen (Rashi).