Quasigryph mount is very typical mount for Alttmers of Clan Direnni. Indriks are revered animals in the Summerset isles, that said getting ones trust to ride on is extremally rare in lore. And of course horse will work as well.
Mostly horses, but as was said before Indriks work too. I would also like to mention that the Atronach mounts found in the crown crates are probably the most lore friendly mounts you can find in the crown crates, especially for the magically inclined races, like Altmer and Bretons.
This Bosmer was tortured to death. There is nothing left to be done.
Pretty much just horses. Other mounts found in ESO are either not as common or not as lore-appropriate as you might think.
Indriks are a good example of an animal which are simply impossible to tame, and so are impossible to mount. The closest thing we have to a tamed indrik in lore would be Ayrenn's indrik in the menagerie, but if you've played that questline you'll know its still a wild beast.
So horses are a pretty good bet for virtually every region of Tamriel, save for Hammerfell which will have some camels thrown in, Morrowind with their Guar, Elsweyr where they will have some Senche-rahts thrown in (though these are sentient beings, not merely animals), and possibly the lizard mounts of Black Marsh.
But when in doubt, go with a horse. You can never go wrong with a good horse.