What does that mean? Protected under what laws?
"only this company can decide who can use the trademark"
It's like:
"only if it's production fulfills certain criteria can you use the name"
It's made as a consumer protection so that people can not just claim their product is this kind of "traditional" product when it isn't. Except that it's less used like a consumer protection as it's mainly used like a trademark. Controlled by a small group of people and directly profiting some regional government/economy.
Most commonly it requires the main ingredient to come from a specific region (also e.g. used for Wine or Sect).
EDIT: I kinda did throw different regulations into one explanation here.
Real cheddar is full of flavour, slightly sour. Great stuff.
But you can go to places like the US and cheddar is not cheddar and they should not be allowed to call that cheddar.
Unless you live in Russia :)
https://en.wikipedia.org/wiki/Geographical_indications_and_t...
Protected under EU law. Only tomatoes of a specific type AND from the specific region can use the name "San Marzanos".
Outside the EU I don't think there is anything stopping yoou from labling any random tomato as "San Marzano".
[0] https://en.wikipedia.org/wiki/Protected_designation_of_origi...
Unfortunately, there is a large black market in mis-labeled San Marzanos, so it's not a full guarantee of quality. But it's something that they're at least working on, rather than just giving up.