Nope most definitely not getting it confused because I‘ve tried various options for the people I shared my details with. I think it had 3 options: Name only, photo only, both.As I say, it’s never worked for me since it was first introduced, maybe back in iOS 13? I do wonder if you are getting it confused with the Edit option under the main messages list rather than an individual one? Anyway, thanks for checking.
Edit: i’ve reported it via the feedback app anyway FB8673032
Not 100% certain though but it popped up a few days ago.