Starring with oneself is always stimulating -- ! but I think in this instance Duke was the horse. Of course, Wayne became "The Duke" to many fans. But this was later.
Actually, and surprisingly, the nickname pre-dates the horse, and it was from his dog!
John Wayne was born Marion Robert Morrison in Winterset, Iowa in 1907, but his parents soon decided they wanted Robert for their next son's name, and changed his middle name to Mitchell (one wonders if he would have been as big a star as "Marion Mitchell Morrison"!). His family moved to Glendale, California in 1911, and there he had a huge dog named Duke, and people started calling him that as well (and the nickname stuck, and he would later name his movie horse that, and eventually everyone would refer to him that way).