I haven't heard of an ethnic Protestant or ethnic Catholic, so wondering what constitutes an ethnic "non-religious" Jew?
The reason, I think, that you haven’t heard of ethnic Catholics and ethnic Protestants is that Catholicism and the Protestant traditions are creedal and confessional by nature. Being a Catholic or Protestant involves subscribing to a list of beliefs. On the other hand, being a Jew is not a matter of subscribing to a creed or confession.
Jews are a people. (Being part of a people is a matter of birth or naturalization and assimilation.) This is how it has been understood for ages. In Exodus, for instance, the God of Israel did not create an Israelite religion and ask people all over the world to become Israelites, rather, he made a covenant with a nation, a people who had already existed—Israel. Jews are just one tribe within the Israelite nation. Even when Jews ignored the covenant, which you could say was not practicing Judaism, they were still called Judah by God and the prophets.
Can one be an "ethnic Muslim?"
There are some South Slavs who call themselves ethnic Muslims. In my opinion, the idea of Muslim as an ethnic label is weird because a Muslim is defined as a ‘submitter’ as in one who submits to God within the context of Islam. If a person does not subscribe to Islam but speaks Turkish, for example, and identifies with Turkish art, architecture, customs, achievements, and the like, maybe that person should just be called a Turk instead of something weird such as ‘ethnic Muslim?’