Because most domestic cats are NOT mixed breed, they are direct descendents of wild cats. Breeds in cats are a very recent (ie. within the last 100 years for a very few breeds, most breeds within the last 30-50 years) development, and most cats do not have any breed at all. Unlike dogs, which have been utilised by humans for thousands of years to do different jobs and therefore have been selectively bred to many different shapes. The only job that humans have expected of a cat is to catch rodents, and they do that very effectively without having to be bred to be shorter/longer/taller etc. Most unregistered dogs are mixed breed. Most unregistered cats are not any breed, they are just cats, directly descended from their wild ancestors, with no selective breeding, no breeds in their ancestry. Which is fairly cool. Your average domestic no-breed housecat is a friendly version of its wild ancestors, like a cat version of a tame wolf or coyote. Wolves and coyotes don't have 'breeds', and neither do most of our furry purry friends.
Those few that are of a certain breed have been created or conserved for aesthetic reasons within the last 50 years or so. Even older breeds like Siamese or Korat basically just originated from ordinary street cats in particular parts of the world, but now bred to each other, to preserve a certain look and personality.
DSH/DLH just means a cat as nature intended. Which is way cooler than trying to make out it's some created breed that just happened recently. That cat in the shelter wanting a home is a mini tiger that wants to curl up on your lap and purr. There's never any need to claim that it's anything else IMO, because the reality is way cooler than anything you could invent.