In earlier times, getting pets small tattoos as a form of ID was actually pretty common - back before microchips where around. Usually it was just a very short set of numbers or letters, usually put in a spot like the inner thigh or ear on a dog for example. You can register your dogs tattoo code into a directory the same way you can register microchips. There are still some people who use these types of tattoos as a form of ID for their pets.
Breeders of certain animals still use tattoos regularly to ID animals as well. Actually, one of my rabbits has a tattoo in his ear. I rescued him and he had it when I got him. Rabbit breeders and showers commonly place tattoos in the inner ear as a form of ID. I'm not positive, but I think it might be a requirement for show rabbits.
Some animal shelters actually give animals a small tattoo, usually in the ear or on the stomach/groin, when they neuter or spay the animal. That way, it's easier to tell that the animal is fixed if it's ever picked up again. Especially for females, it can be hard to tell whether an animal is fixed or not without using more expensive/invasive procedures such as an ultrasound or exploratory surgery. When my sister found a stray dog, we knew she was already spayed because she had a green line tattooed on her stomach - which is what the local shelter does when they spay females in her town.
I'm not saying that I support tattooing all over your pets body like in some of those pictures - but I can see it's uses as a form of ID. When I was debating whether or not I wanted to get my dog microchipped, I actually considered getting him tattooed as an alternative to a chip. I don't see too much of a difference personally between inserting a device under my pets skin, and a very small tattoo. Some vets will give the tattoo while your pet is already sedated for a neuter or other procedure.
Here's a picture of my Barnaby's tattoo.
It's on the inside of his ear. It's not really noticeable unless you already know it's there and look for it.