But it’s not parmigiano reggiano
Yes, it is. Some dumb fuck ultra conservative European laws don't change that. Australian law doesn't currently protect the name, and I can go out right now and buy parmesan cheese that wasn't made in Italy.
The law changing would make it illegal to keep doing that, but it wouldn't change reality. Reality being that the type of cheese they sell today is the same as the type of cheese they're selling in the future.
edit:
Let me put it this way: parmesan is a type of cheese. Think of it like "milk chocolate". I could buy milk chocolate made in Switzerland, or in Belgium, or the UK, or here in Australia. Differences in ingredients and the sources of those ingredients would all affect the ultimate taste and texture of the chocolate. But they don't change the fact that it is chocolate. There's no good reason that cheese and wine should be any different, except for nationalism and conservatism. If you want to call it "parmesan made in northern Italy", be my guest. And that should be protected, because "made in X" is a clear descriptor, and you shouldn't be allowed to lie about things like that. But the name of the product itself should not allow some conservative fucks half a world away to have special privileges. And a so-called "free trade" agreement shouldn't be honouring those strict restrictions on the act of free trade.