Judaism is a both system of religious beliefs and the unifying cultural expression of a group of people (which some would say is the basis for nationality, that's also complicated). The mix of these two things varies within various strains of Judaism. And some, myself included, consider themselves Jews for cultural/historic reasons but not for religious ones (I am a materialist and a Marxist so the big guy in sky with a white beard thing doesn't work for me).
Gentiles have had difficulty dealing with that duality. Sometimes cultural Jews are ok but religious ones are not ok, and vice-versa.
Another complicated variable is the question of Zionism, which many Jews currently subscribe to but is a relatively modern phenomenon. The eastern European Jewish socialists united in the Bund were not Zionists and fought against it as a political trend in the Jewish community. Unfortunately the Bund was basically wiped out in the Holocaust.
I am sure there are others on the list who can speak with more authority on these issues but these are just a couple of thoughts.
Alan Jacobson