While humans are members of the animal kingdom, our powers of recognition, specifically related to the sense we use, is primarily related to visual stimulus and to a much lesser degree, our sense of hearing. In this, man is unlike most animals, as eye sight is used far less often for the majority of animals, and even when it is used, usually it isn't given the same importance that man give it. Let's look at the basic five senses and some of the ways animals use them to recognize each other.
Smell: The sense of smell is one of the most often used in the animal kingdom for recognition. Ants leave a scent trail with which other ants can recognize a member of the colony (though they also use touch). Moths and many other insects use pheromones to attract members of the opposite sex; the scent being a very powerful stimulus. A large number of adult mammals use scent to recognize their own young, even when they are with a lot of other young, and the babies likewise can recognize their parents in the same way. Many pack animals, such as wolves, mark their territories, not only to warn off others of the same species, but to identify themselves to other members of the pack. Dogs often identify each other by sniffing one another, and even cats do this. Cats also 'scratch' objects such as trees, not just for visual clues, but because their feet have scent glands that produce identifying odors that other cats can recognize.
Hearing: Many animals have acute senses of hearing and are able to discern very slight fluctuations in generated sound, which they use to identify and recognize each other. This is especially used by the more advanced animals. For instance, wolves often hunt when they cannot see each other and recognize their pack mates by the barks and howls they make. A mother bear will hear and recognize the cries of their young from a distance, and will also recognize if the call is one signaling distress. Whales such as humpbacks, blues, and sperm whales, use calls to identify themselves and to recognize their pods or groups. Many birds recognize not only their own species, but their mates and young by the calls they make.
Touch: Many insects, such as ants, recognize each other by touch, usually from antennae. However, while touch is used in many other animals as an aid to recognition, it is not as commonly used as other senses.
Sight: Humans, and many primates, use eyesight for identification. Some insects even do this, such as bees. Some creatures of the ocean, particularly of the deep ocean, might not even be able to breed and perpetuate their species if it was not for being able to 'flash' light in the deep darkness, in ways that can identify them to other members of their species. But a large number of animals have relatively poor eyesight; for example bears and rhinos. They must rely on other senses, and they do so very successfully.
Taste: Though at first thought, this sense wouldn't be used much for recognition, it is actually used by a large number of animals. Certainly one cat licking another is 'tasting' the other cat. Many insects use this sense, often combining it with touch. The fact is that a great number of animals use tongue grooming as a form of identification, as well as for other reasons.
It would be a mistake to think that animals primarily use one sense to recognize each other, and another mistake would be to think that they don't use all their senses in one way or another for this identification. It is true that for any given animal, there is usually a primary sense that comes into play, but almost always, this is backed up by other senses, depending on the animal and the circumstance. As an example, wolves clustered together in a pack may be perfectly comfortable with using primarily eyesight to recognize each other. But on a hunt, in the darkness of night, in thick forest, when they can't see one another, eyesight isn't nearly as useful as hearing.
We are still learning some of the wondrous ways that animals identify and recognize each other, and the studies are continuing. It is safe to say, though, that animals have often found very ingenious ways to make the recognition possible.