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)?
There is a kinder variant for such - prosecco, which can actually mean OK quality (within sparkling stuff bounds, which are not high).
But yeah calling anything Champagne apart from stuff coming from Champagne is a bad form and looked down upon a bit. Small kids do such mistakes and thats fine, adults shouldn't.
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.