We can guess ages of cats and dogs based on their teeth, but it's always a guess. In general, permanent teeth begin to erupt at 11 to 12 weeks of age for felines and canines. By the time they are 4 months old, most felines and canines have all of their permanent teeth. When a cat or dog is older, veterinarians look at the condition of the teeth to try to determine age. But this is, as I said, a guess. Just like people, some animals have a biological disposition to the formation of tartar and the development of gum disease. By the same token, a dog or cat might have had a dental cleaning, making their teeth appear younger than they actually are.
With horses, you can actually judge age nearly perfectly based on the teeth and the pattern of wear that emerges as they age.
With chinchillas, their teeth grow continuously throughout their lifetimes. This is why chewing is so important to them! It helps keep them filed down.