Humans and other animals ‘pee’ or ‘piss’. But I only hear ‘piddle’ associated with dogs. Why is that?
I’ve heard it used with small children. And if you’ve heard it with dogs, I bet it’s with small breeds. I don’t think a St. Bernard piddles.
Horses piss. Children pee. Toddlers and Chihuahuas piddle.
Just heard it in relation to a chocolate lab on The Dog Whisperer.
Not all dog pee is piddling. When you have to let the dog out in the morning to take care of business, that’s just pee. But when the dog meets a bigger or more dominant dog and gets nervous and dampens the floor, that’s piddle. And since the size a dog thinks it is is only loosely correlated with how big it actually is, yes, even St. Bernards can piddle.
I once had a male lab, and every three or four yards, on a walk, he would do just a few drops of pee. This is a territory scent-marking behavior, I believe. It is piddling because it is a little bit (a piddling amount, no less) all over the place, rather than one big impressive stream of piss, and it is a characteristic behavior of male dogs. It may look a bit feeble, but actually it is fairly aggressive behavior, if you are a dog and you understand these things.*
So yes, dogs (bitches, not so much) do piddle, but I am sure I have heard the word used of others too (including toddlers, as Colibri noted, because, they often just do not make all that much water).
*On the internet, nobody knows if you are a dog. 
Pee-mail.
Pee-male.
How much piddle would a poodle piddle if a poodle could piddle poodle piddle?
My Chihuahua pisses gleefully. On everything he is allowed to get close enough to - plant, tree, telephones, you name it - and a fire-plug at our street corner, of course, more-so after being ‘fixed’. After a few minutes of walking, his leg-lifting has nothing left to come out, but darn it, he’s gotta try and continues the entire ~20-30 minute duration! Silliest dog I’ve ever known.
I’ve always associated the word piddle with fooling/messing-around with something ~aimlessly, like “piddling around out in garden”, or maybe piddling meaning ‘not really accomplishing much’ - not urine-related. Maybe I should drop my pants when I tell wife I gonna go piddle around outside? 
I have heard piddle used for tiny bits of urine of any origin, but that was very long ago.
Years ago I visited acquaintances in Puddletown, formerly Piddletown, which lay on the River Piddle, a truly piddling little river.
[Sondheim]
What’s the muddle in the middle?
That’s the puddle where the poodle did the piddle.
[/Sondheim]