If so...how would we know? Some horses appear to have split personalities - but these are often easily explained. For example, a horse that gets anxious at a show may be upset by the difference in routine or picking up on their rider's stage fright.
I admit I have known horses that definitely had different "modes" - I know one right now who loves jumping most of the time and then has days when she Just Doesn't Want To Do It, and we're still trying to work out what's going on in her cute little grey head. When she's in one of her moods she won't even willingly step over a pole on the ground.
But in every case of equine "split personality" there has turned out to be a reason - a difference in the rider, a new horse being introduced, even in some cases the weather (I have known many horses that get very "up" and want to run if it's windy).
There's no indication that horses have true multiple personality disorder. Just dramatic changes in mood that can usually be put down to something.