There's a long history of discourse within Judaic studies, history, sociology, and so on, on the multi-level way Jewishness manifests as a social role and identity. It can be a Religion, Ethnicity, A nation (Zionism broadly, as in the Jews need a homeland, or more narrowly---that homeland must be located in current Palestine), or some have even argued a race.

I'm not going to comment on the validity of all these things, simply remark that it is complicated to at least some degree. There is a long cultural tradition of secular jews wondering where they fit into the picture given the kinds of contradictions that arise from the above overlapping but non-synonymous senses of identity. A Jew may self-identify as culturally a Jew, but not be practicing the faith. They will attend holidays and engage in rituals as a cultural practice, not an explicitly religious one. By way of analogy, many Americans and Europeans celebrate Christmas as a cultural practice, not because they are devout christians. Alternatively, a person may not identify as a Jew themselves, but because of their ancestry or whatever they may be ascribed Jewishness---much like poc are when they have mixed parents---as a social identity imposed on them by outside society. Much like the socially constructed fact of race in general, this conception could be considered a race, but that is imposed from the outside rather than in. I'd argue if Jews start identifying as a race themselves that is a reactionary tendency, as any 'race' conception is a reactionary concept so far as I'm concerned.

Most orgs self-identifying as Jewish are RELIGIOUS orgs. They're one of the few civil institutions that can survive and replicate themselves under the grinding erosion of capitalism. Only the most devout stay involved because they are willing to live, practice, and participate in the group despite the social and material costs of doing so. Go figure, zealots are willing to commit to the bit, and that bit tends to be reactionary. Jews who are culturally Jewish but not religious likely participate at the fringes in this sense; that is their community and family whether they like it or not. However, they almost definitionally won't be rising to positions of power because they are themselves marginalized within their own community.

This is honestly a sad phenomenon where people who by all rights should be able to consider themselves culturally Jewish are denied their own identity by a hostile and reactionary in-group trying to impose conformity from within, but they are ALSO put upon by conflation like you seem to be doing in your comment. We are encouraged to simply not be Jews---which is not so simple---for guilt by association.