I have full physical custody but we share legal custody.
He has changed contact info through the school and doctor regarding contact information.
So instead of me getting a call to pick her up from school, he is contacted . Even though I’m the one going to pick her up because she lives with me.
Also doctors bills were not being sent to me because he changed the information to his info. So I wasn’t receiving anything until the bills were stacked up when she had an appointment and couldn’t attend due to a balance owed.

Is he allowed to change this information without my consent?

Hi, I'm not sure of the legalities, but I see no harm in having both of your contact details listed if you share legal custody/there's no court orders prohibiting or limiting this? I'd definitely be writing to the school, explaining how she lives at your address-Are you both 'next of kin?' Is she your daughter (this isn't stated in your post)?