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.
More comprehensive solution here
I think your only option is to contact support for that. I think I remember reading that it is possible with their help.