We can and we do. It's just that some product names are limited by origin - like a piece of (other) cultural art, food also has a locational history.
But I do agree that maybe average customer don't care about that.
But I do agree that maybe average customer don't care about that.
Indeed, even the headline of this article about P-R just calls it "parmesan", which is the same non-DOP term used to refer to similar-style cheese produced in many other places (eg Hungary) and sold in the USA.