To add to this, it is actually forbidden to have a different main residency than the one on your Personalausweis if you live in Germany. If you move, you have to change it within a few weeks.
If you have residency permit, that gets updated. If you don't have (EU citizen) then having done the registration is enough. (I think GPs wording isn't particularly precise: what matters is the registration - and if you have a German ID document, then that gets updated at that point)
