Put simply: Fuzziness is a real thing.
Fuzziness not only in the topic here but in the meta discussion of the topic.
Yeah, there's something about white, American Christians to be discussed historically… boiling it down to "white" is probably better done by going with "European" but that's probably my own intellectual biases from my experiences of the language.
The thing about being Jewish is that it simply has been significantly different (even as there are parallels) to most other ethnic discussions. There are genetic similarities (read: inbreeding) within the primary Jewish ethnicities where other groups show more diversity. There's a weird tie between culture/religion and ethnicity.
I often make the analogy to being Cherokee or something. You don't just decide to believe some religious things and it makes you Cherokee, but they do have traditional beliefs and the idea of true adoption as a Cherokee isn't crazy. The Jewish religion is sorta like that where they discourage converts, and conversion is like getting adopted by a different ethnic group. That's quite different from Christianity and Islam that promote conversion.
For this and many other reasons, Jews stayed distinct in their own weird way.
Talking about "whiteness" and "white people" is actually more like "blackness" and "black people" where dark skin can come from a range of cultural and ethnic backgrounds. Lumping all dark-skinned people together makes little sense except for how it's treated (self-fulfilling prophecy somewhat) in our world in reality.
You may recall people remarking that our first "black" president was not only mixed-race but his African father was really African and not a descendant of American slaves. But obviously he still lived his life in a way that had him be a part of the American (slave ancestry) cultural reality.
So, yeah, fuzziness. But Jews really have had ties going back generations that the various conglomerations of "white" Americans have not… even as you can draw some analogies and questions.