What precisely is a Jew? That is a sincere question.
I have been studying the Word - Old and New Testaments - since age five. I was attending pastoral conferences by special invitation starting from age sixteen.
I learned at age eleven that I was part Semite on my mother’s side (via a woman of real Hebrew lineage, Hebrew name and Hebrew surname). I was pleasantly surprised: I had spent so much time studying the Israelites that I was glad to know I shared some kinship with them. (It was not a big deal in our Christian family - which was my mother did not make much of it.)
I studied Hebrew at a public university. I was the lone Gentile. Only the teacher (a native of Israel) and I knew the Old Testament to any significant degree. This was entirely germane: As a formal language without a country for two millennia, Hebrew became a stagnant language. Modern Hebrew is very close to ancient Hebrew. For that reason, this teacher, though she had degrees in Hebrew and English, often used Old Testament passages to teach usage in context. I usually understood those contexts; the “Jews” in the class usually did not.
So when I say they were ignorant, I know this to be true; yet the rest of the class were all “Jews”: secular, hedonist, atheist, as far as I could glean. Each had gone through Bar Mitzvah or Bas Mitzvah - but it obviously meant little.
They were not hostile to me, but they were not accepting either (although the teacher to some extent was). Many of them had spent time in Kibbutzim in Israel. They were very aware and proud that they were so-called Jews: It was very much an us-against-them environment.
I have wondered ever since on what basis they are Jews?
They say it is not a race.
They say it is not an ethnicity (not really, since they are culled from any and all ethnicities).
They say it is not a religion (not really, since so many of them denounce Judaism).
What is it, except an excuse to think they are better than everyone else?
The closest I have seen to a valid answer was a post recently (where I do not recall, perhaps here): that they are a modern socio-political construction, and are not the “Judahites” of the Biblical times.
I now find the use of the term, Jew, to be rather absurd than valid in today’s social, religious, and political contexts.
Here's a sincere answer: a Jew is anyone either born to a Jewish mother or who has converted to Judaism Halakhically according to Orthodox Jewish law.
The word you are looking for is “tribe” with all its religious and socials aspects.
One is Jewish if one’s mother is Jewish. It’s not a voluntary status. (One is also Jewish if one converts, and I’ve found as a lifetime of being a Jewish Israeli is that Jewish converts, being voluntary adherents, are overwhelmingly more seriously Jewish than those born Jewish.)
The social aspects of being Jewish are as profound as the religious, in that they were designed by G-d to make us a people apart, whether we like it or not.
Some born-Jewish people don’t adhere to the religious aspects of the Tribe.