IANAL, but 99% sure that the situation in Germany is that you have to identify yourself (give name, birthday, (address?)), but you /don‘t/ have to carry (or show) your government issued ID at all times.
I worked through large parts of PQP ([1]) with only the prerequisites that you state, as part of a Uni course taught by Aleks Kissinger.
I got quite comfortable with the formalism and could solve most problems without too much issue, but at the end I still knew nothing at all about quantum physics. YMMV