Thing about [PET_EXOTIC] in native DF, is that they're only applied to cave creatures. The dungeon master is a dwarf who has dedicated his life to understanding the creatures that dwell beneath the earth, along with whatever precious metals and such might appear down there. He is an underground-centric trainer, so there is no reason why he would know how to deal with an alligator or a harpy. He just understands naked mole dogs, and loves them for their oh-so-wrinkly skin. Mmm, wrinkles...
*Ahem*... Anyways, there also lies the problem in that not all animals are really all that intelligent, and if they are they might not want to be trained. Dogs are rather intelligent creatures, and they are very keen on doing what their leader wants. As such, they are perfect candidates for training purposes.
Cats, while they may be intelligent, have absolutely no inclination to do what you want them to do. Attempting to train a cat is a lengthy and dreadful process that can never truly be called successful. Trying to get them to charge into combat or sneak along beside you in the hunt would be a futile effort.
Horses are just dumb. They can, with significant training, be conditioned to not follow their own thoughts. War horses are not called war horses because they charge into battle and start stomping stuff, they are called war horses because they have been conditioned to not panic or change their course under the chaotic stress of war. A perfectly-trained horse would only ever do what its rider told it to do, and would always do that. There are exceptions, of course, but I've only seen one horse that will pull a beer out of the fridge for you. Also, horse are not particularly stealthy creatures, and are only of use in a hunt when you're in an all-out chase.
Bears. They are independent, and thus not instinctually tuned towards following something else's orders. They are also only moderately intelligent, but they can be taught simple tricks. I don't care how much war bears and bear cavalry units rampage through the internet, they're just not likely.
War cows. I'm not even going to go there.
Raccoons. Worse than cats.
Pikes, carp, sturgeons, lampreys, and sharks. Something that can still perform most of its standard actions without the need for its head should not really be considered intelligent.
Foul blendec. It can't even decide if it's undead or not, how would it be able to learn complex commands?