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I think I used to pretend I was a dog a lot when I was a kid... in some ways I still kind of regret not being a dog. A horse... not so much. They are glorious animals, but unless I were a horse living entirely outside the tyranny of man, I don't think I'd choose that particular animal.
But as to the question... Why a horse? Because they are beautiful and elegant, because they are bigger than humans, which instills some kind of respect/envy evidently (one can look to early metachat posts regarding this subject), and because they have a certain power of speed. The unicorn is just modified horse-legend, as is the Pegasus; I don't even count them as different things. Mankind has always been in awe (in one way or another) of certain animals: the horse, the lion (or other big cat), and the snake perhaps principal among them.
it's not necessarily true that you could turn into a horse (according to modern phyics). most people think "the mind" is emergent, but there's no real proof that something like a soul doesn't exist. so it could be that the best that can happen (if there is a soul, and it is somehow removed from physical laws) is that you turn into a horse that still thinks it's human (rather than a "real" horse that thinks, if it thinks anything at all, that it's a horse).