I know for most people this sounds stupid but i would love it if you would be able to interact more with your horse.
I think it would add even more immersion for some people if you could implement things like:
Feeding the horse in towns
Able to to give the horse apples or oats.
Leading the horse on foot in rough terrain.
Grooming, hoofcare, Rubbing it dry with straw after raining.
Let it drink out barrels, a spring in the forest…