Your horse has the ability to play certain emote animations at various times
Emotes your horse can perform include rearing, grazing, bucking, digging, scratching, laying down, whinnying, spooking, and more!
There are a few ways to force your horse to emote on command. Each way offers a few different possible emotes for your horse.
From your Riding HUD (top center of your screen while sitting on your horse):
Use the icons on the HUD to spook, buck, or rear with your horse, or to pat your horse's neck
From the horse's left click pop-up menu:
Select Animations > Posing to lay on your horse's back, or to lay down and cuddle your horse
Select Animations > Action to make the horse buck, whinny, dig, nicker, rear, scratch, spook, and stomp
These animations (in the Action menu) are animations your horse will play while wandering. Not all actions will animate the rider to match.