Yes, you can bring your dog or cat with you to Mexico. Basically, there is a form that you can review in any Mexican consulate before you come, depending on your country of origin. The basic requirement is that the dog or cat must have all their shots or the paper that says it’s been vaccinated and the paper that says who’s the owner. You get these from a vet.
Sometimes the dog or cat owner can even get a permit right away at the Mexican consulate, if they follow the requirements. If they don’t get the permit right away, they go to the vet, make sure that the pet is up to date on all their vaccinations, and provide information on the pet. There are some vets that even put a picture, the name, the address, and the pet ID and just bring it to the airport or drive across.
(Dog being ridden by a toy cowboy, San Miguel de Allende, Mexico, pictured.)