That is a very good question. It is very wide application in nature, going far beyond the question of dogs and humans.

For a start, land vertebrates all have the same general layout of heads, with the eyes paired and with the same relationship to nose, mouth, ears and so on. Both within the same species and between species, it frequently is a matter of life or death, not to mention prosperity or reproduction, to be able to recognise that general structure, so it is not surprising that the necessary software is innate (in-built or instinctive, if you like) in most species. We need it to recognise others, to recognise threats, objects of attention, opportunities, social circumstances such as dominance, just sit down and think about how often you, or a dog or a cat or a parrot behave according to the faces and eyes you see and way you turn them and what they are doing.

Next time you see people watching animals in zoos keep an eye open for when they interact. You will find that most species are nervous of direct stares. In nature a direct stare is often a threat. There have been hundreds of millions of years of selection against vertebrates that could not recognise a direct stare, whether from a member of the same species or any other.

The innate recognition generally does not depend on recognising precisely the face of the same species; that not only would be a tremendous burden on the information capacity of the genome, but would be less valuable than being able to recognise faces in general, while learning meaning of the subtleties of the faces of one's own species as one grows up. The amount of flexibility that we gain in this way is tremendous. However it does not extend to totally different structures, as one can tell from the intellectually and imaginatively stunted creativity (or lack of creativity) that we see in science fiction stories and films.

Now, all that had to deal with vertebrates. Some years ago I was shocked to see that it extended to at least some invertebrates as well. Where I live we have a smallish species of wild chameleon. Various flies commonly are its main prey. I am extremely fond of chameleons and often have carried them around, hunting flies. Particularly welcome species are some of the large, succulent blowflies. Normally a chameleon swivels its eyes in independent directions, but as soon as it spots a promising item of prey, it moves its head around until both eyes come to bear, then it takes aim and proceeds to dine. However, though this works fairly consistently with smaller species of flies, such as houseflies, I repeatedly have seen blowflies, that had been sitting very much at their ease as we approached, including as the chameleon turned its head, hunch down abruptly as soon as both eyes came to bear. It was then fully alert, and often able to take off in time to dodge the tongue.

There is a good deal more in that observation than the question of feeding chameleons.

Just remember that the innate facial recognition of other species works both ways. A human doesen't have to be taught parts of an animal's face such as: these are the eyes, these are the ears, this is the mouth. We just know what is what. I wouldn't humans to be any better at this skill than the typical dog.