You'll still be allowed to put tags in your guinea pigs' and rabbits' ears, but no more tattooing or piercing unless it's for the animal's benefit.
Similarly, the bill provides that "No person shall tattoo or cause to have tattooed a companion animal unless such tattoo: (a) is done in conjunction with a medical procedure for the benefit of the companion animal and to indicate that such medical procedure has been done, … or (b) is done for the purpose of identification of the companion animal" in connection with an official "animal tattoo identification registry." (In either case the bill specifically bans tattoos "for design purposes," and I invite suggestions as to what the hell that means.)
Pet Piercing Will Soon Be Illegal in New York [Lowering the Bar]
(Image: Rascal with tattoos, Lars Plougmann, CC-BY-SA)