I grew up in an area where a lot of people bred race horses and I went to school with quite a few people who looked after horses. So hopefully some of this information will help.
Horses can sleep on their side, but because they're flight animals (as in fight or flight), they have to feel really safe before they'll lay on their sides. This is because in the wild, if a predator came across the horse, the time the animal took to get back onto its hooves could potentially cost it its life.
Horses will sleep standing up as more of a defensive measure, where ligaments, tendons and muscles in the legs lock together. This also has the effect of reducing the muscle function and relaxes the muscles in the legs.
Now, the thing is, horses need to lay down in order to achieve REM sleep.
So, Taurs could potentially sleep standing up, but if they were human at some stage, they would need to sleep on their side in order to achieve REM sleep and probably more often than horses. Otherwise, you're quickly getting into sleep deprivation.