There are many laws, including the
14th Amendment to the US Constitution, that cover equal treatment of people. Horses are not people, and do not enjoy the same rights and legal protections we do. In fact, horses are often
ground up into food and
adhesives.
I am currently an officer of the "
"He-Man Women Hater's Club." As a private club, I may deny you and anyone else admittance to said club, for pretty much any reason - I don't like your hair; you have the wrong color eyes; it's Tuesday. My behavior may be unfair, or even discriminatory, but it's not illegal*, and you may pound sand.
*Note that certain establishments, like restaurants, do come under legal scrutiny because they have an interaction with the public. There are many other examples where discriminatory behavior is prohibited, but a private horse registry ain't one of them.