There was a book about this that used to be available on Prime Reading: https://www.amazon.com/Real-Food-Youre-Eating-About/dp/16162...

Balsamic Vinegar got a chapter, IIRC.

From my memory of reading Real Food, Fake Food, they don't care about the term "parmesan"--that term has been rendered generic, much like "champagne" in the US. But "Parmigiano Reggiano" is a protected term with a PDO ("protected designation of origin"). That's the one they care about.

[0]https://www.amazon.com/Real-Food-Youre-Eating-About/dp/16162...