Well is it? Naively one might think that every animal would have unique wrinkles to their skin, but ... do they?
It appears that no-one really knows for certain. But grubbing around with Google I have discovered:
The nose print of a dog is as unique as a fingerprint, and your dog can be positively identified the same way. Reference.
It is known that gorillas and other primates do have fingerprints, of special interest however, is that our closest relative, the chimpanzee does not. Koala bears also have fingerprints. Individual fingerprints appear to be restricted to humans and gorillas. Reference.
US scientists and criminal justice investigators have developed a technique designed to more accurately track and conduct a census of some animals. The research focuses on the fisher, a member of the weasel family and the only carnivore known to develop fingerprints. Reference.
The only reference I can find to cats' pawprints is this, which sounds like a school project.
But then are human fingerprints actually unique?
It is often assumed, but has never really been proven, that fingerprints are unique, in humans or other animals. The history of this apparently involves an assertion (early in the 20th century, as I recall) that they were unique, this assertion was accepted by a court, and they've been pretty much never really been analyzed thoroughly beyond that. (It's not clear to me how you'd go about proving it anyway, since the pattern of fingerprints for any individual is a function of his environment during gestation (yes, identical twins do have different fingerprints..). So the best you could hope to do is to prove the odds of an interference are vanishingly small. Reference.
Which is worryingly true. Human fingerprints have never been subjected to scientific and forensic scrutiny in the way that DNA profiling has been. This article in The Register summarises a New Scientist report (hidden behind a paywall) of an official report. Conclusion: fingerprints have never been scientifically scrutinised properly.
As for cats ... Well in their usual inscrutable way, only they know!
Oh and here's today's piece of gratuitous pornography. :-)