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
Worth looking into any COSS (commercial open source) companies that you find interesting.
That’s where you write the code that then others glue together - plus you can inspect the codebase beforehand!