(How) should I describe the company within my bachelor's thesis?
or should I go into more detail like e.g. the history of the company?


No, in general, there's no need for that: your thesis is not a company's brochure. Provide enough information to present the context of your thesis, but don't add too much unnecessary details.

Anyway, I suggest you to discuss this also with your university supervisor: they surely know what is the policy or the tradition about this in the university, and they can act as buffer between possible company's requests and university requirements.