In this example, the company is called Bayerische Motoren Werke AG. BMW is one of their trademarks.
See also how there's a company named 'Dr. Ing. h.c. F. Porsche AG' with a brand called 'Porsche'. But nowadays that company is just a subsidiary of Volkswagen, and they could restructure to make the Porsche AG disappear, without doing any changes to the brand.
But they're not examples of mundane descriptions. BMW is much closer than Volkswagen is -- I doubt anyone has ever actually referred to a car as being "a car of the people" -- but "Bavarian motor works" is not a phrase that you'd expect anyone to use descriptively unless there was an appropriate referent, a set of car factories in Bavaria whose company affiliation was obscure or irrelevant. (Perhaps because there are many of them affiliated with different companies, but we're talking about things they have in common such as a demand for steel or technicians.)
Had that been the case, and had the phrase been in common use, it would have been impossible to trademark "Bavarian Motor Works" to refer to part of the Bavarian motor works. But reality is different.
