Champagne is absolutely not a "generic" product in the US.

While some California wines were and still are sold as "California Champagne", there was a court decision back in 2006 which affirmed the same protected trademark for Champagne as in the rest of the world, with the caveat that wineries already using the "Champagne" name would be grandfathered in and permitted to continue doing so.

A bottle of Champagne made in the US will be labeled with its origin, and no one can start a winery and start selling Champagne in the US. That's not the same thing as "generic".