IMC:
Dwarven Pets
In the Red Hills, dwarves favor hounds, dire boars, and birds of prey that will scout for them when they leave their subterranean shelters. They also keep herds of sheeps and goats, but those are strictly for meat, never pets.
In the mountain range known as the Rostfergs, chamois (sort of mountain goats) and mules make good beasts of burdens, and trusty animal companions.
Dwarves living in the Crown, the circular mountain range ridging the Inner Sea, prize eagles as pets. They also have dwarf hounds (see CC1 or CCR) and some breed of subterranean lizards. Finally, they often befriend giant eagles, and they bred griffons.
In the Royres mountains, the dwarves like falcons and hawks. They also domesticate a few felldrakes, and they have the same dwarf hounds as their neighbours from the Crown.
Elven Pets
Grey elves favor dolphins, orcas, and seals. They also like seagulls and Crows'-Nest-Dragons (dragonnets from an old dragon magazine). The largest ships provide accomodation for letting a giant eagle or even a roc land. Lucky enough captains sometimes have one of these wonderful beasts onboard, and they provide valuable scouting and navigating potential and fast travel toward other ships.
High elves like big, nasty attack dogs, and graceful, fast horses. They bred fiendish cats and fiendish snakes as pets for the decadent pleasure of feeding them mice and watching the vain struggle and agony of the rodent.
Wood elves live with wolves and dire wolves. Some tribes are able to befriend bears or boars, sometimes. Special breeds of elks or deers are sometimes used as mounts.
Gnome Pets
Forest gnomes live with all sorts of foxes -- dire foxes, blur foxes (cousins of the blink dogs), giant foxes (which are to normal foxes what giant eagles and giant owls are to normal birds)... But they are known to befriend nearly every animals living in the forest, so you can encounter anything in a forest gnome village.
Rock gnomes have a predilection for burrowing mammals, like foxes or badgers or even rats, but can be found with any other critter mankind has domesticated as well. Like the halflings, they'll adopt large dogs to facilitate their safe travels.
Night gnomes have an affinity for nocturnal mammals and birds, like owls, cats, nightingale, and so on. They sometime travel atop giant owls.
Svirfneblins have few pets in their dark caves. A few bats. Sometimes a befriended behir. Mostly, rather than having attack animals and pack animals, they'll rely on earth elementals.
Goblin Pets
Goblins are famous for their wolfriding skills. They also like any animal that look scary enough -- dire geckos, dire bats, constrictor snakes, krenshars, kamadans, displacer beasts, tazelwurms... Although they can't be considered real pets, vermin will be used to guard certain area, or as means of execution.
Hobgoblins like any creature that can serve as an endurant steed, most notably the horses. Bugbears tend to use the goblin word for "food" more than one for "pet" when discussing animals, but they like wolverines and bears.
Halfling Pets
Lightfoot are famous for their numerous breeds of dogs. They also have domesticated a breed of large (well, Small, in D&D technical terms) wildcats (crossed with lynxes long ago), and some breeds of monkeys, which they got from...
... The ghostwise halflings, who have bred monkeys for several purposes -- scouting, message-carrying, filchering, and even trap-disarming. They also domesticated some snakes and selectively bred them for cleverness (Tiny to medium vipers, with Int 2, harmless to most everyone as they cannot inject their poison into a prey they haven't swallowed whole already, like the Montpellier snake, they are rear-fanged).
Tallfellows and deepstouts halflings have the same pets as lightfoor halflings and, respectively, wood elves and crown dwarves.
Kobold Pets
Kobolds like pseudodragons a lot. Pseudodragons do not necessarily like kobolds, which often have a sadistic streak that displease the dragonnets, but on the other hand, they are provided with a share of the loot and all the fawning care they may want, and then some. They also like all sorts of reptilian and scaly critters, and have stolen and bred for themselves the halflings' smartsnakes breed. Sometimes, they use trained and captured destrachans, other times they'll use dire weasels or large vermins like monstrous centipedes.
The flying subrace of kobolds known as urds are one of the only races that have successfully befriended the loathed yrthaks.
Orcish Pets
Orcs like the ferocity of boars, wolves, and wolverine. They also like the toughness of buffaloes and muskhorns (CC) and similar beasts. All sorts of elephants and mammoths, when used on a battlefield, brings tears of joy to an orc's heart -- such a magnificent strength! such a colossal power of destruction! such a pair of large tusks! -- even if the pachyderms are used by their enemy.
