The Legend of Zelda Tears of the Kingdom is a direct sequel to The Legend of Zelda Breath of the Wild. Assuming a player has a Breath of the Wild save game on their Switch (or in the cloud) does anything transfer to Tears of the Kingdom?
Yes, registered horses can be transferred over to Tears of the Kingdom. You will need to visit a stable first, however.
Note that when you first show up at a stable, if you're not actually boarding a horse, it won't seem to have worked, as they'll make a big to-do about how you're a new customer, and need a new loyalty card. You need to actually show up with a horse to board, at which point, they'll tell you that you can access your mounts from BotW.
Also, the save needs to be downloaded to your console. Being in the cloud is not enough.
Whether that is the only thing that transfers over, or whether there is more, I can't say.