To Add: I am also not entirely sure that all cowboys did originate in Europe, as I'm certain that there were people from all over the world coming to America and becoming "cowboys". I would also like to add that I am Canadian - so my perspective on Cowboys is what I watch/read/hear - that said - I do recognize that not EVERY American is a Cowboy - just that it seems that all "olden days" Cowboys are American.
The distinguishing factor is likely that the United States had a "Wild West" which was merchandised on a global scale. In Luis Francia's History of the Philippines, José Rizal takes inspiration for a nationalist movement from the "dignity...of the American Indian performers" in Buffalo Bill's Wild West show.
Buffalo Bill found a way to sell the "Wild West", by doing shows with cowboys and Indians, horses, cattle, etc. and taking those shows all over the world.
The early 1900s also saw the beginnings of globalized media, and American Cowboy/Indian stuff was easy to merchandise to children. Jon Gnarr, previous mayor of Reykjavik, mentions in his memoir that he read a ton of comics about American cowboys, but that he felt more like an Indian himself. He gets in trouble as a kid for building a literal campfire in his room, so he can read his Donald Duck comics by the fire, "just like the Indians did". But I digress.
Again, the cowboy image that we have today of ten-gallon hats, lassos, chaps and yeehaws, was merchandised well enough that adults and children alike thought it was all a very real depiction of what was going on in the United States at the end of the 19th, beginning of the 20th century.