You are basically given a better starting position because of the association with the name Feta, Champagne or Cognac.
If Nike, the North face or some other trademark owner moves production to a different factory, you can’t continue production of the same goods even if the quality is the same.
The GI and trademark name translates to extra value.
Because that is what the "feta-like" phrasing is doing in consumer's minds. Put the same feta side by side, where one is labeled "DOCG greek feta" or whatever, and the other is called "feta-like cheese", and the greek feta will outsell the other 100% of the time.
Further, Feta, for instance, has existed for hundreds of years. It's no more owned by one person than Greeks themselves at this point. It's spread over the world, in the same way as Parmesan, or other cheeses and foods. The supposition that this is exactly like trademark law is absurd and mechanical.