Is it ethnicity? Religion? Nationality? In some ways, it is all three.

And more! It's cultural, political, genealogical, religious, ethnic...a whole mish-mosh.

Even has a whole wiki entry:

Wikipedia: Who is a Jew? - Wikipedia

Most of Israel's neighbors just want to see Israel gone. I don't think they care about Jews living in Florida (or anywhere that isn't where they live).

I suspect that's not universally true. I mean, they might not care about Jews living in Miami. But all of the Jewish communities that existed all throughout North Africa and the Middle East were destroyed in the 1940's and 1950's. Those were very large populations, I should point out - Iraq had one of the largest Jewish communities in the world after WWII, and many Ottoman and North African nations had received the Jews that were expelled from Spain back in the 15th Century. As has been pointed out a few times, about half of Israelis are descended from immigrants from those Muslim countries. If Israel becomes "gone," I don't think Iraq is going to allow the hundreds of thousands of descendants of the Iraqi Jewish community that was driven out between 1948-1953 to return. There is now a widespread (though also not universal) rejection of Jews being allowed to remain in Muslim countries.