Having a generic brand name doesn't mean other companies can do that, as there is no legal term defining generic brand names (i.e. a brand name that became synonymous with a product).
Actually, it means exactly that-- as you can tell by the number of companies selling "aspirin" or "cellophane" or "linoleum".
Trademark is precisely the protection that prevents others from selling a product with your name (or one so similar as to be easily confused.)