Well it's a common question I've asked of people during my life, and only catholics ever make that distinction, almost a correction. And there's a reason why - because they think it's the one true church.
I'm not in the USA btw.
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.