Well with the example of a boar tusk, no matter how you look at it leather armour is infinitly better than just bare skin, or hide. Most armour is not meant to withstand the force of a hit. This is because the force would simply travel through the armour into you and break something.
Instead most armours aim to deflect. A broadhead arrow that grazes by a patch of bare skin will leave a pretty bad gash, if you had leather over that area, it would get cut up a bit, but your skin is saved and you would not bleed.
Anyhow what if we went with the idea of multiple breeds.
You have your average dog. Good as a companion and can deal some suprising damage in a fight. Pretty much the dog we have now. You can train the dog for hunting or for war. They breed pretty quickly so they are more or less expendable.
Later on in the game, you might import some dwarven war canines or tame some similar wild animals. Instead of being like your average dog these are more akin to giant wolves. More than big enough for a dwarf to ride on. Because they are large, strong and slightly more feral you have to take FAR more care in raising and training each dog. Again, the big investment.
Once the dog is ready for war, you would want to protect that investment. This means adding some heavy armour, ESPECIALLY if you intend to use them as a mount. There is nothing strange about dwarven canine cavalry.
Full dog armour would not be an everyday thing. It would be fully customized and time consuming to make but we hardly lack for materials. Partial dog armour would be more common and is simpler to make.
So the different levels:
Large War Dog mount: Full plate, this is as much for the rider as the mount. Cavalry gets shot at, a lot, plus they need the heavy armour when they crash through enemy ranks and recieve many hits as they go by.
Large War Dog zerger: For these you would use a plate helmet and chainmail "blanket". The helmet protects the dogs head and eyes. A blind dog will not do much fighting, and the head and mouth are the most likely to be injured. The blanket hangs over the dog and protects it from missiles. Since these dogs are quite strong they can bear the weight easily.
Plain War Dog: Metal helmet and leather armour. These form the bulk of your "rampaging mob" of dogs. They are not as valuable as the specialized breed but you still want them to get to the enemy and do some damage. Enemies tend to shoot at the most intimidating target and a flood of war dogs certainly fits the bill.
You would defenitly want to put a metal helmet on the dog for the same reason as the large war dog. But chainmail would tire the dogs out too quickly and slow them down, they are not as strong and fast as the bigger breed. Leather armour would server to protect them from glancing missile hits.
Hunting dogs are built entirely for speed. The only armour that would be practical is leather and even then a full doggy suit would limit movement. A helmet and a "vest" would be more than enough.
A skilled leatherworker can form the material to pretty much any shape, armour fit for dogs would be even more simple than armour meant for humanoids. A proper dog saddle would take a lot of effort though.
Chainmail is a bit harder to work with but there are no complex joints to worry about, just a rectangular blanket that sits on the dog like a cloak and a couple straps to make sure it does not fall off.
When you get into the area of plate armour, the cost skyrockets. A full suit covering the head, neck and body (not the legs) would have to be made in seperate parts, all tailored to articulate properly when put on the dog.
On the other hand a plate helmet for a dog is nothing special, just a piece of sheet metal curved to fit the dogs head with eyeholes. These could be made for any dog in the fortress without much trouble.
So that pretty much covers all levels of sense.
Plate armour + pet dog = no
Plate armour + gigantic dog mount = YES!
Dogs + incoming bolts + simple armour = good chance of angry live dogs