Oh i dunno, i wouldnt discount it completetly.
Dogs at the moment are certainly considered expendable. Training a dog seems to take like... 10 seconds, even with a rank novice trainer and POOF you have a war dog! instant double damage!.
Well in reality a properly trained war dog first needs a strong dog. This takes the right breeding and a lot of food. Then you have to train the dog, probably from adeolescence on.
It makes sense that you would want the dog to be as effective as possible, that includes defense as well as offence. An optimal trained war dog represents a HUGE investment, something you would not want to lose to one lucky hit.
The same thing applies to hunting dogs, only the training is even more intensive, instead of just attacking you also want the dog to distract the animal your are hunting. And before you even begin, you need the dog to FIND animal in the first place, a war dog could not do this.
So, large investment. Added to the monetary value, a pet that you live with for a long time pretty much becomes a member of the family.
All these factors make it pretty clear you would not want to lose dog.
That brings up the good question, what do we armour the dog WITH? Assuming we have the same options as we have for dwarves:
Leather is a good option, hardened and perhaps studded leather is more than enough to ward off the blows you would expect to recieve from wild animals, effectively giving the animal a second, tougher skin. Dogs fit for fighting are quite strong and would barely notice the weight of leather.
Chainmail is the next option, very tough and can be formed around any shape. Chainmail is weaker against crushing and piercing then it is against slashing damage, but overall it is much better than leather.
Plate armour is probably overkill. But again considering the investment and the danger the dog will face plate armour is really not overdoing it. There are a few problems however. Even the strongest dog will be slowed considerably by the weight and would not be suitable for long pursuits. In tight tunnel combat the heavy protection the armour affords is put to good use.
Even if a "full suit" would be rather much, head, neck and chest protection would be enough to protect the animal from most threats. But if the animals start coming under missile fire then all-around protection is warranted.
All of this applies to other animals suitable for combat. Eleplants, bear, great cats. Crocodiles maybe? Armoured dragons and hydrae would be night unstoppable.
Coming back to cats. I agree that cat armour sounds rather ridiculous. But the largest of tomcats get rather huge and rival most dogs in terms of strength. The problem is the more you domesticate the cat the more huggable they become and the less predatory they are.
Cats can still be trained, circus cats are a good example. In Dwarf Fortress cats act as antivermin exterminators but also attack other creatures fearlessly.
So lets say, you took one of those large cats and trained them to attack vermin. The variety of creatures in DF is hardly limited to bugs and mice. Because of the danger involved cat armour really doesnt look like such a bad idea.
All-around armour would not work, but a cat would need protection for many critical areas.
The foremost of these is the head, but the head protection must be sparing as it cannot limit the cats periferal vision and must allow the ears to swivel.
The chest and upper body would need some cladding because these are the areas where the cat would most likely take a hit.
The spine of the cat could do with some protection, but said protection must be highly flexible or it will interfere with the cats movement.
The tail of cat might not strike you as a critical area, but it is worth protecting simply because it is a big and vulnerable target.
Plate armour would be waaaay overdoing it. For one thing the cat relies on its agility, plate armour articulated enough not to limit movement is scale mail and something else entirely. the weight would also upset the cats finely tuned balance and limit its acrobatic potetial.
This would be less of an issue when it comes to big cats, a giant panther for example would easily bear the weight of some iron plates. Plates made of adamantine would be light enough for even your pet cat.
Chainmail is good because it has unmatched flexibility. A cat with a short "hoodie" of chainmail is much more protected and loses none of its capabilities. The only problem is that to distribute the weight of the maille, you need to attach it using as many points as possible. Im sure you can imagine the temperment of a cat festooned with straps and buckles, if it could move at all.
Leather is perhaps the best option, it is light and can be tooled to fit any shape. Leather armour would fit the cat like a second skin.
So you see? its really not that ridiculous when you stop to think about it. Any discreptancies can be worked around or are rendered null by the setting of the game.
This is dwarf fortress remember. A cat that lives in a cave and hunts for its own food is a far cry from that fuzzi pile of fur monopolizing your pillow. Likewise for the dogs, a daschund makes a grand companion, but a sled dog he is not.