To be ridden, an animal must be strong enough in the back and large enough to carry the usually male warrior. They must be tractable and responsive, both. Also, they need to be speedy and enduring.

The first cavalry animal was the domestic donkey, shown pulling a chariot in the Royal Standard of Ur. Only later were horses imported from the Steppes, as "donkeys of the mountains" since they had to come over the mountains from the plains. (Clutton-Brock, The Horse in History)

Cattle are too slow to use in war. Dogs and llamas were not big & strong enough. Dromedary camels fit the bill, with several cavalry breeds, but Bactrians were not bred that way. Moose have been tried, but while usable as messengers won't stand for battle. Ditto reindeer/caribou.

You see, first comes domestication, then comes cavalry. No one ever said "Let's domesticate lions for saddle." They would look around at what they had and see which ones they could ride and later which ones they could ride to war. Horses were first domesticated by the Sredni Stog steppe farmers, probably for milk as they had nothing like cattle or sheep, just pigs for meat.

Horses seem to have developed the basic traits, which were furthered by selective breeding. The traditional warhorse, from Xenophon's On Horsemanship through Blundell's Renaissance manual, is always a stallion, which is a pretty brave, aggressive & dangerous beast, but rarely inclined to kill his rider, & not for lunch. It is more sensitive than a donkey, mule, or hinny so it takes cues from rein & leg better, which it must to be controlled when the rider is fighting. Horses are loyal and game, and will keep going even when mortally wounded. They will do the impossible for a rider they trust.

Elephants were used in war, but they are so large & slow that they only work as mobile platforms.