It seems that Horses are not the only thing Link can ride in the Zelda: Breath Of The Wild as he can also mount and ride Elks as well. YouTuber SwimmingBird941 has recorded footage of doing just that and also a lot of other things you can do in the game. It is worth noting that even though you can befriend and sooth the elks much like the horses, they do not sprint or able to be registered and kept in the various stables dotted around Hyrule. There may be somewhere in the game where you can store them but that remains to be seen.
If you would like to see the video created by SwimmingBird941 and see how Link can wrangle the elk and ride him, we have provided it below. You can find the scene at 2:10