Child apple id cannot be used to create an account

We have the same problem. My son can’t sign in to his apps to save his progress (under 13). I’ve talked to someone and they said it was the developer, but it can’t be because that would mean every developer did the same thing.

You can make an Apple ID for your children

In iOS 16 or iPadOS 16 or later

1. Go to Settings > Family. 
2. Tap the Add Member button 
3. Tap Create Child Account, then tap Continue.
4. Enter the child's name and birth date. Be sure to enter the correct birth date — you can't change it later.
5. Follow the onscreen instructions to finish setting up the account. For your child's Apple ID, you can use their email address, the suggested @icloud.com address, or their Game Center nickname. 

In iOS 15 or iPadOS 15 or earlier

1. Go to Settings.
2. Tap your name, then tap Family Sharing. 
3. Tap Add Member.
4. Tap Create an Account for a Child, then tap Continue.
5. Follow the onscreen instructions to finish setting up the account. You can use the child's email address for their Apple ID, or their Game Center nickname. Be sure to enter the correct birth date — you can't change it later.