This is because the waters have been, seemingly intentionally, muddied. I have seen criticism of Judaism, oddly enough even criticism by ethnic Jews, as being considered "anti-Semitic" and similar terms. Furthermore, the distinction in my experience seems to be rejected by many Jews who consider "being Jewish" as being essentially and intrinsically related to the Jewish religion enshrined in cultural practice.
I also don't think that ethnic group really captures it either.
For me, being Jewish is just its own thing, sui generis, that only partially overlaps with these other concepts (religion, ethnicity, nationality, etc) but isn't any of those things.
A counter example would be the American designation of 'Latino' as an ethnicity, independent of race, language, and culture, and based entirely on national origin.
I recommend reading The Invention of the Jewish People, by Shlomo Sand.
As a practical example. When my father (Ashkenazi) married my mother (Sephardic), my father's grandma was devastated, because he was marrying a non-jew.
Except for the genetics we share, distinct from non-Jews, statistically significantly.
I recommend reading "Legacy: A Genetic History of the Jewish People" by Harry Ostrer.
We really are the people of the book, aren't we?
> A Jew from Yemen has nothing else in common with a Jew from Poland
1) They descended from the same people 2) If they are religious, they have a tremendous amount in common
From the Catholic perspective, it is primarily a theological identity, as evidenced by the way the New Testament uses the word. Jesus, his mother Mary, the Apostles, the first so-called Christians, were all "Jews", and so in some sense, we can speak of the Jews who accepted Christ as the Messiah, and those Jews who did not. The New Testament describes the latter as "the Jews", and since there is no ethnic difference between the former and the latter, it follows that "Jew" in this context is not an ethnic, but a theological concept[0]. This isn't to say there is no ethnic sense, but the theological concept is the real focus.
Among Jews, Jewishness is taken to be communicated matrilineally, and the interpretation of such phrases as "children of Abraham" (or "seed of Abraham") is understood in carnal terms. Most Jews today are not observant, yet they still view themselves as Jews and are viewed as Jews even by observant Jews. Cultural Jewishness might seem like one way to be able to maintain Jewish identity without the theology, but because Jewish culture in the traditional sense is so theological (and frankly, all cult-ure ultimately is), one has to ask what kind of Jewish culture are we talking about? Once the underpinning and sustaining beliefs are gone, the culture loses its force and meaning, even if it takes a while for the old attachments to wither to completion. Some new theology would need to take the place of the old while maintaining the old name. Without the old theological basis, carnal identity begins to take on an even greater importance.
I might also touch on the civilizational definition of "Jewish", though because civilization is ultimately determined by religion, the question is ultimately a question of theology. Koneczny's definition comes to mind.
So the word is overloaded. But generally speaking, we are very much living in a period of identity crises. Some claim that the uptick in racial politics is the result of the decline of religious identity in the West.
[0] Incidentally, this is why some of the accusations of antisemitism that are made of Catholicism are wrongheaded and involve some kind of modernist projection of categories alien to Catholicism. "Antisemitism" is a term coined in the 19th century (conventionally attributed to Wilhelm Marr) that has in mind a racial and ethnic meaning of "Jew", but the conflict between Catholicism and Judaism is not ethnic or racial, but theological. The Rabbinic Judaism that emerged after the Crucifixion, with the destruction of the Second Temple that left the Jews with no temple sacrifice and no priesthood, is taken to be defection from the Faith, something analogous to Protestantism (recall that the Old Testament is a history of such defections, and is the history of heretical Christian sects). Judaism is taken to have transformed from a faith centered on sacrifice to one centered on observing the law. The sacrifice and the priesthood are taken to continue in the Church--the New Israel--and in the mass, now bloodless and perfected[1].
Wishing for more Jews to become much more aware of their ethnicity would be one of those monkey's paw wishes.
The analogy I often give others, is that we are more like Australia's Indigenous peoples rather than Christians. An Indigenous person doesn't need to believe that the Rainbow Serpent created the world, they are still Indigenous. Similarly, a Jew doesn't have to believe in the Exodus or anything from the Tanach - they are still Jewish.