DeLuca named NACDA Athletics Director of the Year Without getting into a much larger debate about what activities are sports and what are games (thank you Ken Purdy!), the "sport" aspect of equestrian is both the athletic ability of horse riding in general, but also the ability to communicate to/with the horse by body position shifts, rein position, etc to get the horse to do complicated things.
Just like good race drivers are fast in just about any car, good riders are good on totally unfamiliar horses.