What happens when the name of the product _is_ the name of the region though. For example, Champagne or Parmesan? Giving your product another name clearly disadvantages you.
In the case of champagne, I'm usually unwilling to pay for the regional "brand." Instead, I often buy "cava," which is another regional AOC for sparkling wine, this one run by producers in Catalonia (and elsewhere in Spain). It's a bit cheaper, which is probably less than ideal for the producers, but I still know more or less what I'm getting.
You could argue that one or the other of these two situations is less than optimal. But as a customer, I find that AOCs and DOCs generally help me buy products I'll like. And since it would be hard for the producers of "Tennessee Whiskey" to enforce their agreed-upon quality standards on a company in New Zealand, it does make a certain amount of sense to limit participation geographically.
What if nobody bothers making quality "Parmigiano-Reggiano" in the US because they know it is close to impossible to compete against the brand name and the margin are not worth it to even try? Maybe, if Americans were allowed to call theirs "Parmigiano-Reggiano" they'd put a lot more effort into making it high quality given that they'd have a much bigger chance of making a profit.
Champagne is already limited to a region in Europe, but that hasn't stopped, say, Catalonia from making Cava: their own regional brand for what is, in essence, the same thing. If anything, this also allowed local producers to band against the old name, which now indicates it's a product made in France.
So in practice, its not a big deal for good competitors.
That’s what’s done here. You see that it is not the exact original product (sometimes you don’t even want the original product, but a specific fake of it), and you see still what it’s similar to.
One that's currently being litigated in the EU is whether labeling products not from Greece as "Greek yogurt" is misleading. I personally think yes, especially given the capital 'G' and the parallels with other clear origin-designating terms like "Greek wine" and "Greek olive oil" (and mixed origin/style terms like "Scotch whisky"), but it could end up going either way.