Who's a Jew?
First off, one of the benefits of Judaism is that there is no central authority governing rules, membership, or definitions. While each denomination, jurisdiction, or governing body has rules, those rules are not universal. Even the traditional definition of "you're Jewish if your mother is Jewish" is not universal; it's a practical custom, accepted by some, that applies less and less in the current era.
I have evolved a lot over the years on this topic. When there is no definitive rulebook, anyone who truly identifies as Jewish can theoretically be Jewish. That's to say, you can be Jewish by faith, by ethnicity, by culture, or by choice. You can be of mixed heritage and still be Jewish. If you're half Jewish but feel Jewish, you're Jewish. And the converse is that, if you are ethnically Jewish but don't identify, you're still technically Jewish, but the next generations will be less and less so (and your right to speak for the community is dicey at best).
Frankly, if you say, "I am Jewish," no one can tell you that you're wrong. We have no pope or president to say otherwise.
And what do Jews look like? Many Jews are truly observant but most today are not. Many Jews are ethnically and culturally Jewish (whether Ashkenazi, Sephardi, Mizrahi, Ethiopian, mixed, or otherwise). Many Jews are connected to the Jewish community in some form, but many are not. For some, the Holocaust and persecution are the main connections. For others, it is Israel. For me, it's a combination of culture (especially Yiddish culture and history) and Israel in all of its profound diversity (Middle Eastern, Ashkenazi, Jewish, Arabic, Druze, African, history, food, music, synagogues, mosques, and on and on).
So, if Jewish peoplehood is so diverse and inclusive, can't a Jew be a proud Jew and still not recognize Israel as the Jewish home?
I think the answer is yes, though I passionately disagree. Our historical, cultural, ethnic, and religious connections to Israel are profound; our need for security is equally so. But some people simply aren't connected to Israel and don't have the firsthand experience that convinces them that it is important. I'll accept that those people can act in good faith, and that their political views on Israel/Palestine are simply political views and not about their Jewish identity.
However, many take it further. Many non-Zionists actively crusade against Israel's right to exist (beyond criticizing its policies or actions) and become anti-Zionists. These people are activists who have generally adopted an extreme progressive narrative about Israel and the "nakba," and who place Palestinian liberation above any Jewish needs. It is difficult for me to accept that those who support self-determination for another tribe but not for their own are truly proud of their identity.
There is yet another group that is even more common among those I know in my life: those progressive Jews who identify (often proudly) as Jewish, are not connected to Israel, and who turn a blind eye to racism against their own people because of their misgivings about Israel or their own identity. They may not be activists like the group above, but they will readily support Palestinian rights before their own and rarely comment on hatred leveled against Jews. At best, they are ignorant; at worst, they are ashamed of their own identity.