Follow-up as I think I’ve confirmed the bug and how to reproduce it.
This issue seems to happen only when the hide_profile profile setting is enabled — then the birthday field on the edit profile seems to just not work as expected (appears to not persist). While, if hide_profile is disabled, then the previously saved birthday shows up. I think this might have been introduced here:
Pinging @zogstrip for now and if no reply, I’ll bring it to Contribute > Bug.