I'm in the UK. Normal conversation: what did you do on Sunday? I was in church. "Oh are you a Christian" "yes". Now, if the first person is not a Christian , that's often the end of the conversation. If first person is also a Christian they would say "oh me too! What sort of church do you go to". They might then answer, oh I got to st Luke's, have you heard of it. Or I go to so and so in the town. The baptist church? Yeah that one" It's very unusual for a non Catholic to go straight to denominations in answer to the "are you a Christian", because that's not the question, and due to the appeal to unity, because of the belief that we're all part of god's family, rather than go straight for dividing lines. And never would anyone identify as "I'm a protestant", that would be odd.
OK thanks. By the UK, do you mean Great Britain? England? Something else?
As far as I know (admittedly not far), Christians in the UK are about one-third Church of England, one-third non-denominational, one-fifth Catholic, and the rest other. I think most of the growth in non-denominational Christian churches in the UK has taken place in my lifetime.
There is a particular part of the UK where, in fairly recent decades, I think self-identifying as "Protestant," as quite specifically opposed to "Catholic," was not at all odd, to use your word. Not sure of the extent to which that's still the case.