You can have 2 stables in 2-player MP, I assume up to 4 in 4-person co-op. And the Horses seem to be interchangeable to the point were we need Hats to tell them apart.
Except it's clear that mine likes to play hide n' seek as it's never where I think it should be. That could also be my MP buddy, though. Come to think of it, that'd be a pretty good way to mess with someone-- just "accidentally" hide their horse behind random trees when they're not looking. Also serves as good in-game motivation to keep Trees chopped down in order to minimize MP shenanigans.