Because they're different species, and because of tradition with registries, clubs, breeds, etc.
In the horse world, are there very specific standards laid out for each type of horse like we have in the dog world?
Quite specific, yes. And generally speaking, those do not end up well, humans tend to do a very poor job of ensuring function as well as form.
Take a look at the AQHA or Arabian Halter horses to understand what is happening there (they aren't the only examples, but two pretty egregious ones). The AQHA halter horse is the primary reason that HYPP entered the breed and was so difficult to extinguish that halter breeders will still breed N/H for the double-muscling that can come with it. Most halter horses are never ridden, and aren't designed to be.
They are completely different than the HUS horses, which are IMO, AQHA in name only. Long history behind that. But breeders don't generally cross Halter horses and HUS horses, any more than you would cross a Doberman to a Beagle. They are, for all intents and purposes, completely separate breeds with completely separate bloodlines and types.
Other breeds have completely closed studbooks, like Thoroughbreds, which also have the requirement of live cover. And their genetic diversity and conformational issues are problematic as well.
I don't agree that they have very different traditions & clubs, the purposes are quite similar. Most of my equine friends are quite involved in the dog world as well. The idea is the same. Conformation, "type", ability, and long term soundness are quite important. I mean, I suppose temperament and rideability are a bit more important in the horse world since a 1200lb animal can literally kill you if it decides it doesn't want to play anymore, but I'd argue that is important in dogs as well.
Keep in mind, I'm not arguing for an open studbook. Please don't get me wrong. I am concerned about the lack of genetic diversity in the breed, and increasingly anxious about the future of the breed as a whole.