Of course, a British Muslim may have nothing in common, ethnically, with other 'British' Muslims. They just believe the same version of nonsense.
Neither Jew nor Muslim is a race. It is merely dependent on which version of nonsense they believe.
I did like that EFL, very nicely put. This scenario always reminds me of the Ohmid Jilali film where he thinks he a Muslim, and then finds he was born a Jew, and then has to come to terms with it.
but Muslims are definitely not limited to one ethnicity
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.