Horses could, and should, be implemented in games like dogs have been in the best scenarios.
Dogmeat in Fallout 3, and the freely-named Mabari in Dragon Age: Origins. I cared for those creatures more than other characters. Although I'm a dog person, and as it tends to stand for many people; the death of a dog, or another animal for that matter, in a movie for example, can be a lot sadder, than the death of a human character.
But still, horses could very well be utilized. And they are practical in the way of transportation, as dogs tend to help you with the fighting, in video games.
In Red Dead Redemption, for the little I played it, I didn't care for the character's horse at all. It was just a tool. The same thing, with even lengthier extent, is true with Skyrim. They weren't even good tools there.
I've ridden horses as a kid and a teenager. My sister owned half of a horse for a good while and another one was often on a field just a stone's throw from my home when I was a young kid. I've carried water for a horse for a couple of Finnish winters. Even still, I've basically never had any kind of connection with horses.
I think the affection for horses is largely different with women and men. I used to work for a couple of summers with two 50-something men. They were kinda rugged, but one time they began talking on horses they, or their families had when they were young, and horses were still used for actual work and they were like family members. They almost got emotional talking about them. How strong, useful they were and what characteristics they had. Whereas all the female relatives, friends and girlfriends I've known, seem to have a very different aspect when it comes to horses.
So, video games still being somewhat dominated by males, and the ones developing them are subjectively young, from urban backgrounds, and designing the games for male audiences... It's not that difficult to see why horses aren't what they really are in video games. There should be a change though.