First, let me state that there are hundreds of types of Cheese being produced in Switzerland. However, only certain types are mass exported. This is handled by the National Cheese Marketing organisation ("Käseunion in German"). They make marketing decision on which cheese they sell internationally. AFAIK that's mostly Gruyère, Appenzell, Tilsiter and possibly Sbrinz (look it up yourself to be really sure). Guess I forget one or two type.

Now historically, Switzerland was very early to produce cheese for more than private consumption. They would produce cheese in the Alps and sell it along the trade routes outside of early Switzerland. This export started as early as the 15th century. When they started to produce cheese outside of the mountain regions in the early 19th century (the first valley cheese maker in German speaking part of Switzerland started in 1815 just across the street from where I type this, in Kiesen, canton of Berne). This did not really take of for 10-20 years, but then cheese production simply exploded and local cheese producers popped up in most villages. Often employed by the local farmer's association.

Being the first international cheese on the market, the brand was just Swiss Cheese. Although I am not entirely sure this is totally true or just a general impression. Also, I do not know a study which looked at the historical marketing of Swiss Cheese. Still, I think Swiss Cheese stands for a certain quality and has done so for a few hundred years. Looking at all the different types of cheese, the ones I quoted above are quite close to one another in taste and form. So if you summarize them as Swiss Cheese, you are not that far off. So this all makes some sense.

However, if actually come to Switzerland and go to a local specialized cheese retailer, you will find a big variety of local cheese. I can have 20-30 types, all produced within 20km.