I would like to make a document stating that my mare belongs to me and another person. What would such a document be called?

I would call it a "Statement of Ownership."

If your horse is registered, the easiest way to go about getting such proof of ownership is to contact the breed association to find out how to get her registered under your name. Usually, you have to submit her papers along with a bill of sale and pay a fee. Then if you and your friend decide to sell her, the new owner can have her registration transferred to his name.

If your mare is not registered, you still can draft a statement on your own. Include as many details as you can: foaling date (if known), age, breed, gender, markings, etc., and include side view and front view photos. Attach the bill of sale, if you still have it. Sign it, date it, have it notarized, and keep it in a safe place, like any other document.

@ Dave B: You must not know much about owning horses. There are some very compelling reasons for documenting proof of ownership, not the least of which is the very real threat of theft. (Yes, horsethieving still happens.) Anyone who boards her horse away from home should be able to establish that the animal does indeed belong to her if, for instance, the facility and all contents are seized by creditors (trust me, this happens too.) I can think of a half-dozen situations in which having proof of ownership makes the difference between losing or retaining the animal in the case of a dispute. "The government" has nothing to do with it.

Get a bill of sale with a detailed description of your horse. Then, go and get her micro chipped by a vet and put those papers together. That should be proof enough for anyone!

Unless your state keeps track of the ownership of horses (which is highly unlikely), then such a document would be called a "worthless piece of paper with no legal significance." The government has no interest in keeping track of the ownership of every bit of banal minutia that rests within our borders. What's the deal, anyway? Are you trying to freeze out somebody else from messing with the horse, or is this some kind of weird declaration of mutual friendship? Maybe you should get one of those sets of neclaces that are each half of a heart that say "best friends for life" or something. Hell, get one that has three parts and give the third piece to the horse.

a bill of sale.... that is what you need.