Hi.

This is not a remnant of Nazi ideology. Freud himself speculated of it in 1934, the idea that Jews are both "racial and cultural" in definition, and that their origins and survival stemmed from the hatred of them, at least in part.

The way you're thinking of being Jewish is purely as following a religion. Judaism is indeed a religion, but being Jewish is more than following said religion. Christian authorities consider you to be Christian if baptized, but if you became an atheist in say, Spain during the Inquisition, you could be tortured or executed, not regarded as Christian any further. Jewish thought holds that you're still Jewish, even if you convert to another religion, because being Jewish is a part of your soul, not something you can give up so easily.

The idea, therefore, that you could be Jewish even if you converted, contributes to this hereditary idea. This was reinforced by Jewish historical experience. There are some exceptions, and there are different strains of historical thought, i.e. adopting paganism vs. converting to Islam, but consider the case of Maimonides, the famous Jewish thinker also referred to as the Rambam.

In 1160 or so, the Rambam wrote Iggeret ha-Shemad, or the Letter on Apostasy. He writes, during a time of persecution of Jews, on the forced conversion to Islam (those who refused might be put to death). He argues that "Shoud one, however, inquire of me: 'Shall I be slain or pro-nounce the Mohammedan confession [and thereby "convert" to Islam],' my answer would be, 'Utter the formula and live!'" He points out that they are required solely to, "recite an empty formula" (the words that would convert people to Islam), and considers this to be nonbinding, and argues that Jews are not even committing a transgression against the faith if they do this to survive and remain as faithful as they can (and seek to leave if they can as well). Already we can see that Jewishness is something innate, not something you can simply leave by virtue of saying some words and joining another tradition. However, the Rambam also discusses keeping Judaism in your heart, which may lead our discussion somewhere else. And obviously, this was not a simple opinion; other Jewish scholars at the same time were writing that Jews could die rather than accept a forced conversion to Islam, on account of it being considered idolatrous by some. But this has roots even in the Jewish discussions of the Old Testament (known as the Tanakh). It is noted regularly that even those who commit horrible, idolatrous acts are still called the "Children of Israel". The Talmud, a body of Jewish law, considers this in Sanhedrin 44a, looking at the case of Achan. Achan is a character who is considered to have sinned in egregious ways. Indeed, the Talmud notes that:

The Gemara asks: Isn’t it obvious that he concealed his circumcision, as Rabbi Ile’a said that he transgressed all five books of the Torah? The Gemara answers: Lest you say that while Achan transgressed all five books of the Torah, with regard to a mitzva relating to his own body, such as circumcision, he did not act irreverently, Rabbi Ile’a teaches us that he sinned concerning this mitzva as well.

So basically, he was considered to have sinned egregiously in virtually every way. Yet the Talmud notes:

When God explained to Joshua the reason for the Jewish people’s defeat at the city of Ai, He said: “Israel has sinned” (Joshua 7:11). Rabbi Abba bar Zavda says: From here it may be inferred that even when the Jewish people have sinned, they are still called “Israel.”

So despite all his sin, he is still Jewish, he is still "Israel". This is the case with children as well; say a woman converts to another religion, and has a child. By most authorities, that child is still considered Jewish. This is something that is well-discussed and well-worn in Jewish tradition. It differentiates itself from other religions in this way, at least in how we understand them in typical fashion.

It's really a difficult thing to understand Jewishness. It is a race, yet unlike races, you can enter it (via the typically difficult process of conversion). It is a religion, yet unlike other religions mostly, you can't leave it. But this is not something that began with the way Nazis racialized Jews.

What is interesting is when people call themselves, for example, "25% Jewish". In Jewish thought, that doesn't exist. In the more historical view of Jewishness, if your mother is Jewish, you are 100% Jewish. That's all. You can be a, let's say, Ethiopian Jew, or an Iraqi one, but you are a Jew. If your mother is not Jewish, you are not a Jew.

In more recent times, some movements have been more willing to accept patrilineal descent, if the child commits themselves to being Jewish. Even so, however, it often is not necessarily an automatic thing, and it requires effort to show that one wishes to be Jewish.

This in some ways differs from previous historical experience, where there appear (according to genetic studies nowadays especially) to have been cases of men leaving the Middle East for Europe, and the wives they married there converted. Genetic studies are still relatively new and hard to parse, though, so further investigation needs to be done to fully understand the founding lines maternally.

“Jewish” refers to an ethnicity as well as a religion. This ethnicity can be traced genetically. The vast majority of religious Jews are also ethnic Jews. A child whose mother is religiously Jewish is religiously Jewish unless they decide not to be. A child is ethnically Jewish if either of their parents are ethnically Jewish, regardless of their religious beliefs. Many Jews see it as more an ethnic and cultural concept than a religious one.