First of all, it is not mandatory for everyone. Anyone with an income above an entry level IT job is exempt, but there are other exceptions as well.

Secondly it is not open to anyone. If you opted out of it at a young age you usually cannot come back (it's a bit more complicated). If you were not in the system for a couple of years and are over 55 you are out. So the US retiree will not get in, but the migrant with a low-incone that worked a couple of years in Germany has mandatory health care forever.

Thirdly, it is not a single state-run insurance company. There are many companies that provide mandatory health care and you can choose freely but you have to choose one if health insurance is mandatory for you. The companies differ in premiums and benefits but usually not by much. They also cannot refuse you if (and usually only if) health care is mandatory for you. The biggest ones are AOK, Barmer and maybe Tk. Bigger companies like BMW run their own ones for their employees.

Of course there are many companies that provide private health care. You can have private insurance in addition or, if you are exempt from mandatory health care, instead of the mandatory providers.

Everything is more complicated, but that is the rough idea.