Europeans definitely try to enforce rules like this worldwide, and AFAIK they're mostly successful, at least in developed nations.

I haven't seen illegitimate Parmagiano Reggiano in the USA. They usually just refer to the US-made version as "parmesan". I also live relatively close to Napa Valley and pretty much nobody here calls wine Champagne unless it's actual Champagne, other than a few companies that still use that loophole I linked to.

You can also make parmigiano reggiano in the US.

I thought parmigiano reggiano was also a protected term/origin in the US. Like Vidalia onions are. Most of the other EU ones aren't though.

That's why "parmesan" is a thing - it's a cheese similar to parmigiano reggiano, but with a shorter minimum aging time, and no requirements on where it's made or what the cows are fed - parmesan can be made with commodity milk anywhere rather than in one part of Italy from a specific breed of cattle fed at least 50% by grass grown in that part of Italy. Other than the aging time the process is similar, which is why the cheese is similar.

This isn’t entirely true, according the article. If a producer in the US was using the name “Champagne” before 2005, they can continue to do so, but producers can’t start using it anymore.

It took two decades of negotiations, but finally, in 2005, the U.S. and the EU reached an agreement. In exchange for easing trade restrictions on wine, the American government agreed that California Champagne, Chablis, Sherry and a half-dozen other ‘semi-generic’ names would no longer appear on domestic wine labels – that is unless a producer was already using one of those names.