Tattooing your cat or dog is okay provided it is for identification purposes. Apparently, too, veterinarians sometimes tattoo a small line on female cats who have been spayed. The purpose, it is said, is to provide a clear sign that the cat has been spayed if the wound has healed so well that it is not obvious.

Identification tattoos are usually placed on the inside of the ear flap or on the inner thigh. These are discrete but visible if a person needs to see it. Although cat and dog identification by tattoo is relatively rare.

PETA says that the tattoo might be the social security number of the cat or dog’s owner. I am not sure about that as a concept. The tattoo should be a number under a database.
As microchipping has become more mainstream tattoo identifications have become less common, I’d argue. The ink may fade and there is a risk that the number may be incorrect as they are very long. Also long numbers are tricky to tattoo on a small animal. In the UK pets tattooed before 3rd July 2011 can be used under the Pet Travel Scheme. Brexit will perhaps affect this.
There was a phase – is it still as popular? – when tattooing cats and dogs was a bit trendy. It lead to bans in New Jersey and New York (in early 2015). Breaking the law can result in a $250 fine and a possible jail sentence. These are specific laws relating to pet tattooing but, of course, where tattooing causes pain it may well and probably does, violate animal welfare laws in all 50 US states under existing general animal welfare statutes.

The New York and New Jersey laws were enacted in reaction to the decorative tattoos on a dog, photos of which were published on Instagram causing outrage. In New York, Assembly member Linda Rosenthal sponsored the legislation, signed by New York state’s Governor Andrew Cuomo. Rosenthal is a great animal advocate as she is also behind the declaw ban.

There have been pictures of tattooed Sphynx cats coming out of Russia. The Russians like their fancy purebred cats and it appears that they don’t consider decorative tattooing in violation of their animal welfare laws. If that is true it may be be because animal tattoos are inked when the pet is under aesthetic. However, there is a risk of infection as I understand it. Also when an animal is placed under these risks (anesthetic also carries a risk) for the self-indulgent pleasure of the owner, it has to be immoral and illegal.

One argument for decorative tattoos is that it deters pet thefts. Yes, agreed but it is a poor argument. Pet thefts are relatively rare and there are other ways to avoid them which are healthier for the animal (i.e. for cats full-time indoor life).
The conclusion with respect to tattooing pets is that it will always be at least immoral to ink your pet decoratively because you like the look of it. Your pet has not given consent for it and it can be detrimental to their health and welfare. Identification tattoos are an altogether different process. They help to protect the animal and the benefit outweighs the small detriment in having ink placed under the skin.
Before 2012 I wrote about ID tattoos of cats – click to read the article.