Adult horses, like most other animals (Caucasian humans are a rare exception), are not normally lactose tolerant (this goes for cats too - please don't give your cat the classic saucer of milk. They will regret it...and so will you).
Adult horses can get diarrhea if given milk or milk powder.
Foals, of course, drink their milk - ideally straight from the source. Hand raised foals are generally taught to drink from a bowl or bucket right away - easier in horses than other animals and less of a pain than bottle feeding. Cow's milk, though, is not very good for foals - it's got too much fat in it and can give the foal diarrhea. Goat's milk is better (assuming mare's milk isn't available). In fact some people have actually managed to convince a goat to nurse a foal, giving the goat a platform to stand from.
Commercial foal milk replacers are available in modern eras.