Back before pet battle and account wide pets, people usually had one fishing alt that they used to fish up magic crawdad, and were able to get multiple 'drops' by fishing. I assume that you can still get multiple crawdads on one toon, the same as the sewer rat, though both are low drop rates, so it might not seem like you could more than one per toon.
As far as having multiples of a certain pet in your pet journal, each pet has a listing whether it is 'unique' or not. Achievement pets, like the jade tentacle and Venus are unique, as well as almost any of the pet store pets, or promotion pets, like the murlocs. Likewise, the only bought pets you can not have more than one of are the guild vendor pets like the armadillo and the dark phoenix hatchling.
Any tradeable pet you can have up to 3 in your journal, and infinite caged. Any quest pet, like the tiny flamefly you can have up to 3 in your journal, and only a few are tradeable, I think fishy is the only one there. Some are odd, in that you can trade it until you learn it, like the red cricket, and mechanical chicken, then they become non tradeable.
TL;DR: Look at the pet pane and it will list if a pet is tradeable and/or unique.