A widespread example of this phenomenon in Poland is calling a particular form factor of shoes "adidas" (plural: "adidasy"), regardless of the brand or manufacturer. In fact, I can't think of another word to refer to this shoe category. Another one is "kserokopiarka" for "photocopier" (and thus "skserować" for "to photocopy"), which derives from Xerox. And I think using "jeep" (pronounced as in English) for a class of cars is also common. And, of course, we Google stuff on the Internet instead of looking it up.

But other than these, I don't recall any popular examples of brands turning into generic nouns in Polish.