While I’m by no means an authority or expert on the subject of Jewish identity, I have talked about the matter with my Jewish wife quite a bit. I have also formed my own opinions, and they aren’t the accepted view among most Jews, but I will do my best to present each in their context and not attribute any of my own views to those of actual Jews.
First, what I am certain of: Jewish identity is complicated by the fact that many views on it do in fact exist in the Jewish community. It’s clearly a religion, but there is much debate about whether non-practicing Jews are “culturally/ethnically Jewish.”
I would simplify the whole thing by making the ethnicity “Hebrew,” while the religion would be “Judaism.” At the very least, the ethnic label should be “Judean,” because the “–ism” implies the practice of the religion, not just a genetic link. I don’t think this will ever happen, given that “Hebrew” caries some negative connotations and “Judean” is eerily similar to the German epithet “Juden.”
But that is also an oversimplification. There are Eastern and Western European Jews, the Ashkenazi and Sephardic, respectively. As is the case with most “races,” there is no real purity, as Jews in the East likely mingled genetically with Turks, while Western Sephardic Jews on the Iberian peninsula have Spanish and Arabic in their bloodlines.
But what makes an ethnicity is arguably culture more than genetics. If this is the case, people who are “genetically Jewish” who were not raised in a Jewish household are not culturally Jewish, and therefore not Jewish. However, there is the small matter of Judaism, the religion, and Israel.
Israel offers a free trip to Israel to all those with Jewish ancestry. In this instance, it is a matter of proving that one has ancestors who are at least partially of “the chosen people.” The criteria is similar if one chose to apply for citizenship in Israel. Also, from the perspective of Jewish proselytizing, there are active efforts taken to get non-practicing Jews to adopt a religious lifestyle, especially on college campuses.
So when someone asks if Judaism is a religion or a race or an ethnicity or whatever, I guess the answer depends on the situation, or more importantly, on the person. I think it’s important to address people by any (serious) label they choose to self-apply, though I find it rarely even comes up. If someone calls themselves Jewish, and they’ve never set foot in a shul, that’s their prerogative.