I have set my new name everywhere I can in the AppleID settings but the old name is still showing up in this place, and maybe in other places which I haven't seen yet. But how can I get my chosen name to show up here?
(for clarity, this is the profile picture selection on iPadOS)
Solution: The solution for me was that I had to change my name in the contacts.
i haven't used apple devices in ages, but if this was an android issue i'd also make sure the way your own contact is saved is updated as well. I mean the way your own number is saved as in your phone. Maybe try that?
This looks like a super annoying "feature" on Apple's side. Did you try calling Apple directly? They are usually quite helpful, even if your AppleCare has expired. Besides I honestly think this is something they should be aware of.
On my iPhone (sorry, my iPad is at work) I tapped on my profile picture in settings, then personal information, then name, entered my name, hit done, then back, then swiped away settings and when I reopened settings and touched on my V profile picture again my name updated.