While I understand what you're saying, even within the EU the name "parmesan" only became protected as recently as 2008. Meanwhile it has been used as a generic name for the style in the rest of the world since...forever. It happens, and that's just the way it is. We sell, and recognize, the protected trade name of "Parmigiano-Reggiano" here in the US.
Even for our own product, Coca-Cola, we do not protect the generic version "cola", in the US or abroad -- this despite the fact that the company and brand, Coca-Cola, popularized and apparently introduced the first "cola" and include it in their name.
Given that, why on earth would we protect the generic term "parmesan"?
I don't really care in the slightest if they want to protect this or that name you can call how you want it for all i care, i was discussing of the fact that they decided to adopt this solution while not having a good substitute name for the product.
The only things that matters is that they find a way to make sure that every customers and not only the somewhat expert ones (and given the food culture in the US that means almost no one) know if they are buying a product that comes from Italy, Greece or whatever or if it's produced locally.
What makes it a fake? Lots of Italian immigrants in North American that used to make cheese in Italy. We're not talking about some complex process here. The ingredients and process are essentially the same.
Fake in the "i'm trying to rip you off making you believe this thing is done in such place and in such a way while in reality is not" way.
Now i'll bring another example in the mix since they were the hot topics of the mini documentary i happen to watch recently.
Like cheese also oil is sold the same way in the US. You see Italian-ish sounding names for the brand with italian flags everywhere and what not when it's actually oil that it's made California. This kind of things deceives the consumer which will tend to think that is buying a product that have certain qualities while it really doesn't.
Also yeah sorry to disappoint you but essentially the same it's not the same here we relies even on stuff like winds to give flavour to the products like ham.
It's exactly like wine, grapes growing only a few Km of distance apart can result in a totally different tasting wine.