Another problem with tattoos is there is no central database. If you find an animal with a tattoo, then what? Do you know where to find the actual owner's information?
Maybe you've heard of tattoo-a-pet, so you might call them but there are many other tattoo registries out there as well so the cat may be registered with a different one. With dogs it's even more difficult because in addition to the pet tattoo databases, dog tattoos might only be registered with the national breed club for their breed, or with the AKC or CKC or other kennel club.
With a microchip, you can usually tell which company the chip is from when it is scanned so you can look up the number quickly and easily.
My dog was already tattooed when I got her, but her breeder made an mistake with her registration papers (her previous owner was sent the wrong dog's papers, with the wrong tattoo number) so I actually have no idea who her tattoo would be listed as being registered to if anyone if someone contacted the right kennel club. When I got her I had her microchipped and the tattoo and microchip both registered with AKC and the microchip company, but there are still a number of other tattoo registries out there she is not registered with.
The other problem with tattoos is beause there is no regulation or central registries, this means anyone can use any number they want for their pet's tattoo. This means your pet's tattoo number could actually be registered as being owned by someone else through other tattoo registries.
Hey I wonder if I could have my pets tattooed with the word "microchip" so people would know to scan them!