Legal requirements for registration of what can reasonably be presumed to be animal breeding and selling businesses is well within the scope of the interstate commerce clause. I don't know the specific law you're referring to, and if it covers people who just have several pets, it's out of line. But the puppy mill problem is not only one of horrible animal cruelty, but also of serious infectious diseases being spread from state to state, since nearly all these operations sell to brokers including out-of-state ones, and the brokers in turn resell to retailers in multiple states. And even for animals only be sold within the state, most states' laws have long provided for regulatory measures to avoid spread of disease in livestock, and commercial inventory pet animals are essentially livestock.
Horses are another type of "pet" animal that are frequently sold across state lines, and have multiple owners during their lifetimes, so breeders/traders of horses can also appropriately be subject to registration and inspection requirements. But of course, this shouldn't apply to a family which keeps a few horses for pleasure riding.
Unfortunately ridesthemiles is not exaggerating; it does.
EVERY "animal premises" is to be registered. It applies to any property on which there is "kept, maintained, cared for...etc." any fowl, poultry (which includes rabbits) equines, ungulates, exotic, etc, etc, with either a single or double toed hoof....
Also covered are fish. Have a farm pond with trout or catfish? Have to register.
If you have a single pet pot bellied pig; a child's pony; a pet duck; a pet rabbit; even Granny's laying hen, means the premises must be registered. No exceptions, no excuses.
Every time the 4-H kid takes an animal to the county fair, I understand, there is a piece of paper to be filed, for a fee, and then filed again to return home. EVERY time ANY animal (so far, pet dogs & cats are not covered) enters or leaves the property for any reason, it must be recorded.
It is a federal mandate, with each state enacting enabling legislation; and, I have have heard, all at the instigation of UN NGO's. There are many threads about it, and several FReepers much more knowledgeable about it than I am.