The origin of the Cowboy lays in Spain. Particularly in the Seville region. But one piece of gear missing from the Spanish Vaquero is a rope and a horn on the saddle. The use a lance or a pole. And have actually developed a dance showcasing the skill of a horse and rider with the lance. The South American Gaucho uses a bolo and not a rope and does not have a horn on his saddle.
So the lariat, lasso, rope was invented or adopted in Mexico and spread to the rest of North America. From what I’ve read, the Spanish may have gotten the idea of using a rope from Native Americans.
Using a rope and a horned saddle is uniquely North American.
Spanish style shown below.