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)
Your name will need to be changed in three places.
Change your Apple ID name here:
https://support.apple.com/en-us/109353
Change your device name here:
https://support.apple.com/guide/iphone/change-the-name-of-your-iphone-iphf256af64f/ios
Change your name in Contacts.
You also will want to log into any email accounts through their web interfaces, go to account settings, and change your name there as well.