TotK has a neat bonus feature that automatically imports your registered horses from BotW - same bond and name as before, all free of charge.
But there are a couple of things to keep in mind about this:
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It only applies to horses from a Normal Mode playthrough. If you only have a Master Mode playthrough on a profile but no Normal Mode files, you don't get anything.
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It is different for each individual profile on your console. If you have registered horses on profile A but start a TotK playthrough on profile B, you won't get anything that is registered on profile A.
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Which horses you get is determined the moment you start a new game in TotK, not when you first interact with a stable and actually get your horses. If you want to get specific horses for free, you have to catch and register them before you even start a new TotK playthrough.
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It also specifically depends on the most recent save file in BotW. Keep that in mind when you mess with the time / date settings of your console (for Animal Crossing, Pokémon games etc.) because the last time you saved the game IRL migth not count as "the most recent save file" in BotW when your console was set to 1990 or something like that.
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The Royal White Stallion and Giant Horse transfer over if you own them. However, if you do NOT import them from BotW, you can still find them in TotK, albeit in new locations. (This ONLY applies if you didn't import them - you can't get duplicates)
Spoiler for the new locations:
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Epona (from amiibo) also transfers over. You're still not allowed to customize her in any way, and you can't raise her stats at Malanya Spring.
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The "default horse" does NOT transfer over. If you have "Link's Default Horse" (unique stats and look) in BotW, then you will get a "fake" variant of it in TotK (slightly different, non-unique look). The original DefaultHorse doesn't seem to exist anymore.
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Equipment doesn't transfer over. You have to unlock the Royal, Traveler, Knight etc. Stable & Bridle-sets again. The good news is that you can now get the Traveler's Saddle and Bridle without an amiibo, and the other sets don't require annoying minigames.
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The Ancient Set however doesn't exist anymore, which is a huge bummer IMHO. It might not have been the prettiest set, but the ability to teleport your horse to your side would have been a LOT *more useful in TotK for the sheer amount of vertical exploration. Even if the saddle only worked on the Surface instead of the Depths and Sky, would it have been super helpful.
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Since Nintendo doesn't seem interested in creating DLC for TotK, it is unlikely that this set will be added unless they do it in an update or via new amiibo figurines.
