Actually, they are, I think we both misunderstood the ethnicity bit, (or at least I did) as it's really another way of saying religion, not race. On the other hand, you could have a sub argument that a Shia Muslim is a different ethnicity to a Sunni Muslim, even though they are both Muslims, just of a different branch.
But Jews will also subdivide in different ways, as you have Orthodox, Reform, and Conservative and Reconstructionist Jews, so you could argue they they have similar sub groups as the Muslims have.
When I said you were a Jew, or Muslim, I meant that that is how they define themselves, simply by their religion being the most paramount aspect of their identity. I'm British, and I was christened in the Greek Orthodox Church, but I don't define myself by my religion, nor by my race (half Greek Cypriot, half English) but by the place I was born.