I guess it's "generic" in the sense that everyone calls a sparkling wine "champagne", same as everyone calls a copier a "Xerox machine", but if some other company would try to sell their copier using the Xerox name, they would still be sued by Xerox (the company)?

The US legal system doesn't have the concept of "trademark" in the way you're using it, that word means something else here.

And no, "trademark" definition aside, it's not "generic" in the US by any definition of "generic" commonly used in the US, legal or otherwise.