“What’s to stop cousins and siblings from marrying, then?” — Hell, I dunno. “Decorum and good taste” is probably out and if they lack socially-normal ingrained abhorrence of inbreeding, they probably are reinforcing recessives already. So the question is really about your right to not be squicked by social deviates; you’re fine if they’ve got to sneak around.
“What’s to stop polygamous marriages? Group marriages?” Bloody-minded prejudice, if you asked the LDS at the right point in history, and that probably motivated by fear of being out-earned and out-bred. Also, these days we have plenty of TV shows demonstrating the pros and cons of one version of polygamy — I wouldn’t sign up for it on a bet. But there’s no rational basis at all to limit wedlock to only two people. And we’ve got lots of divorce lawyers who’d welcome the work. (Man, they’d get fat on it!)
“How about an adult marrying a minor or horse?” (or “…a chair?” etc.) Aha! That, there’s a basis to deny: none of those entities can give meaningful consent or enter into a binding contract.
Now there is what appears to be a concerted effort to lower, or even obliterate, the age of consent; but I suspect the culprits engaged therein are less interested in long-term relationships — children eventually grow up, after all — than in simply getting their hands into some poor kid’s Garanimals.