I think it's important to notice that the point is not to keep how to make the cheese secret. The recipe for the cheese is actually encoded in the law, and you can download it freely ( https://www.parmigianoreggiano.com/it/consorzio-disciplinare..., in italian). During lockdown I was trying to make a regional cheese, and the law was the best recipe I could find, with extremely detailed steps.

I think they are trying to discourage the usage of "something style" naming, as consumers associate the "something" with the knock off. I saw it a lot in recent years in the UK, even knowledgeable people end up buying "parmesan style" cheese and refer to it as parmesan, when the taste is not the same.

To protect the name in general, they might be going after every instance of the "style" usage, even if in this case it's clear there is no intention to trick, and probably no commercial incentive.