Well, Morrowind kind of broke the line for just about any familiar domestic animal - the main game and the first expansion pretty much intentionally avoided any real-world animals in favour of something more exotic, and thus no cats, dogs, pigs, sparrows, etc., just guars, alit, kagouti, cliff racers (
), etc. Oblivion kind of skipped over domestic animals in general, and I guess pigs ended up being missed as well in favour of cows. And Skyrim was going for a Viking appearance, and stereotypically that means more dogs than cats, and plenty of cows but not neccessarily pigs.
Also, there are plenty of cats in Skyrim. They're just bipedal and like this Moon Sugar stuff.