It's rarely "exactly the same". Champagne made in France and Feta made in Greece has specific ingredients, requirements etc, defined by law.
In places where those laws don't apply, they make cheap knockoffs with different ingredients mix, laxer process etc, to sell cheaply. Danish or American feta, for example, is nothing like actual feta. At least Denmark cannot name it "Feta", whereas US companies can, thus misleading the consumer.
Second, even in cases where it's exactly the same, it's a "point of origin" protection, meant to protect the original region that made the product.
The same way that BK cannot name one of their burgers "The Big Mac", but for a whole region/country, not just one private company.
In the USA, you can't sell onions as "Vidalia onions" unless they're a specific species grown in a certain part of Georgia where the soil has very low sulfur content. It produces a sweet onion that you could only get with specially prepared soil anywhere else. https://en.wikipedia.org/wiki/Vidalia_onion So, you can imagine with something like sausage, where there are many ingredients that are subject to local conditions, it would be impossible to regulate all the criteria for being identical to sausage made in that region, and it's easier to just limit by geography.
I've always viewed it as similar to a regional copyright protection in that a region is allowed to advertise that only it produces the "real" product. Much like any shoe manufacturer can create a shoe exactly like Nike, but only Nike can advertise as actually being Nike.
