Less a brand name because multiple companies can make parmigiano reggiano, but it’s a combination of requirements designed to protect local industry - for example, for it to be parmigiano reggiano cheese it has to be made with one of two lists of three ingredients, the milk has to come from cows from a specific region of Italy, a certain percentage of the feed for those cows must come from a specific region of Italy, is aged for a certain minimum time, etc, etc. It’s an entire set of industries protected by a legal definition of a cheese.
When you see “parmesan” instead of “parmigiano reggiano” it’s a similar sort of cheese that isn’t made within the legally protected definition. Most often it’s just not made in the one specific part of Italy with milk from cows from that part of Italy fed by feed from that part of Italy, it’s made somewhere else using dairy that doesn’t have to be imported. Or it’s aged “enough” for the flavors to develop but not the full time required. Or both.
There are a whole array of product designations in the EU that basically exist to protect individual agricultural industries from competition by requiring that products be made in a certain place, or using products from a certain place in order to prevent outsiders from duplicating the product, increasing supply and driving down prices.
Basically the same logic as “if it’s not from the Champagne wine region in France it’s just sparkling white wine.” Also the same reason why “real” balsamic vinegar costs a fucking fortune.