While mounting onto a horse, the player's hand moves like its trying to hit a button. Essentially clicking on a horse makes the game play the Button Hand Press Animation.
Comments 4
Seconded, it seems as though successful right-click actions should universally play a hand animation.
EDIT: MC-164692 has been confirmed as a legitimate issue by mojang due to it being assigned a priority, so I'm even more sure that this is intended.
This is probably intentional as MC-127094 has been fixed.