Statutory health insurances are Körperschaften des öffentlichen Rechts, which (as the name implies) are public institutions and very much not private companies. They have the right to bear seals, can issue titles to collect missing payments, are bound to administration law, etc. It's the same legal designation as e.g. the city of Munich which hopefully nobody calls a "private company". It's really just their marketing which looks more corporate than what we are used to from most other public institutions.

Many hospitals are part of a municipality or a university (again, established by public law), many others are organized as private companies (either publicly or privately owned). Non-hospital doctors are almost completely private entities.