My contacts’ names change to what my child names them

I'm was encountering the same problem, and my child and I are definitely using different Apple IDs from each other. I manage his contacts via the Screen Time feature, and his contacts are now mixed in with mine. Screenshot from Contacts.app on macos:

I found the problem, at least in my case: I had added my child's iCloud email account so I can see their emails in Mail.app. When I did this, their contacts were added to mine and (for some reason) chosen in favor of the names I had assigned to the same people.

The solution was to disable the "Contacts" setting for my kid's iCloud account:

I had the same issue so for any others here’s what I did.

went to accounts and stopped my kids contacts syncing… then eventually removed as I can get his email on his phone anyway.

my name kept showing up incorrectly in family sharing still. Turned out my son’s contact card on his phone had my mobile number. I updated to the correct number and problem was solved.

It sounds as though your sharing an Apple ID. When you do that, the people sharing the ID can make changes in your data. The solution is for everyone to have their own Apple ID.