You can't change the date of birth or family relationship of an Apple ID with a calculated age of less than 13 years old. Otherwise it's supposed to be possible to change the date of birth associated with an Apple ID. I've done it before (for good reasons).
Recently I decided to get #2 child an Apple credit card. Since banks take birth dates seriously I decided I needed to correct his before applying. It didn't work!
This is what his birthday looks like on his Apple ID web page (same as in his iOS devices):
I suspect his account entry is somehow corrupted and possibly mixed with another account. Otherwise I wonder if this is a side-effect of Apple introducing family credit cards; maybe once I got an Apple credit card all the family birth dates were locked as a side-effect.