Technically, they're all mutts since they all were the product of inter-breeding, but as you noted, at some point a mutt is no longer a mutt.
Once people became very interested in owning and showing "purebred" dogs for appearance rather than utility (early 1800's),money obviously entered the picture and the combination of money, prestige, and what can seem to most of us a weird obsession about the minute of breed purity, led to a great number of disputes. Into this chaos stepped regulating organizations. Some previously existed to document and confirm the lineage of dogs bred for hunting and racing, and expanded to address the new "fad" while others developed from scratch. Theses "clubs" were breed specific and they still exist today, but eventually they combined to form larger regulating bodies.
The AKC or American Kennel Club was formed as a registry of purebred pedigrees for dogs bred in the United States, and while it is highly influential, it is not the governing body for the world. In the UK there is The Kennel Club, in France, the Société Centrale Canine, and so on.
There is no international governing body to determine which breeds are "purebred." Each national or regional body decides what breeds it will recognize and accept in their registries. Thus, a breed that is recognized by the Kennel Club in the UK might not be recognized by the AKC.
The Fédération Cynologique Internationale exists to facilitate competition among dogs from member countries, but its rules and breed grouping don't necessarily apply to competitions within individual countries. It's somewhat like the way the IOC operates with sports. If you watch the Olympics, you'll see that the some sports are played differently there than in the US. The FCI "recognizes" breeds, but only for the purpose of allowing them to compete in international competitions.
I have to admit that I am not sure of all the precise criteria for recognition of a new breed, but population and popularity come into play as well as a sufficient degree of distinction from any existing breed. There has to be a large enough number dog of the breed with documented pedigrees going back a set number of generations. Basically the AKC wants to be able to say that an actual distinct breed exists. A breeder can cross breed any two purebred breeds to produce a litter of pups in a new "breed," but that can only be a recognizable (by AKC standards) fact if there have been say 1,500 examples from 6 different bloodlines through 5 generations. There are other criteria that are in place to prevent excessive inter-breeding. You can imagine that establishing the documentation requires an organization dedicated to the would-be new breed (money) and a fair amount of time.
While I'm sure that there are people who are devoted heart and soul to a new breed and long for the ability to have their prize specimen compete in AKC sanctioned show, there is also a lot of money at stake. If a breed isn't recognized by the AKC, an individual dog of that breed can't be AKC Registered.
People tend to think that an AKC registered dog is somehow guaranteed to be a good
dog, and to some extent this is true if your definition of "good" is a dog that to one degree or another meets the set standards for that breed. There is a better chance that your dog will not have severe inbreeding problems if the mother and father are AKC registered, but all purebred breeds have some problems common to their breed because on in-breeding. Still, the AKC registration is seen as a stamp of approval and justifies high prices.
There are currently 180 AKC recognized breeds, with new ones added on a seemingly regular basis.
There are 17 Maintenance Class Breed which is sort of a holding stage for breeds that are seeking full recognition.
And there are 49 Foundation Stock Service Breeds which is a class for the very rare breeds that are not or may not ever be in a position to apply for full recognition. It is the AKC's way of helping breeders of these dog to establish the documentation they need, which is pretty good thing on their part. This is the class where you find dogs most people have never heard of: Estrela Mountain Dog, Catahoula Leopard Dog, and Treeing Tennessee Brindle.
I don't agree with firefly that the crossbreeding "fad" that has produced labradoodles, golden doodles and all sort of "oodles" and "ugs" is going to pass any time soon. People like novelties and to the extent these novel cross-breeds (neither hybrids, nor designer) can at least be said to provide special characteristics, e.g. the temperament of a lab and the (hyped) hypo-allergenic qualities of a poodle, I think they will continue to have a market. The real trend in breeding is in keeping with a move we see elsewhere in the culture: the promotion or development of very rare breeds and the re-establishment of breeds that while they can't be considered extinct have essentially disappeared. This sort of "artisan" breeding though is not likely to have a major impact unless somehow someone resurrects the "perfect" breed.
As for allergies, the majority of people with do allergies are allergic to the dog's dander (dead skin) or saliva. No matter what coat type a breed has they all have dander. One would think that a short or wiry haired breed would be the best choice for allergy sufferers, but shih tzus and yorkies are touted for being "hypoallergenic" and both have long hair. On the other hand, there are short haired dogs like bulldogs and boston terriers that are problematic. In their cases though it is probably because of their excessive slobbering. Also, your dog's hair can attract allergens like dust (or more accurately, dust mite feces) and pollen, so if you are looking for a "hypoallergenic" dog, it's important to know precisely what allergens cause you problems so that you can make a somewhat educated choice.
If you're allergic to dog saliva, you don't want a dog that's always licking you but I don't think any breed is known for excessive or minimal licking, and you won't be able to tell if you're buying a puppy because they all lick. The reason being that it is an instinctive behavior designed to trigger regurgitation of food from their mothers. Think of that the next time your dog licks you.
Finally, I saw an interesting show on TV that said that dogs that run feral and interbreed over a number of generations eventually produce offspring that look strikingly alike, no matter where the populations develop, and your dog is a spitting image for that “breed.” So if your dog isn’t a Chinook, it looks like an example of a “purebred breed” all its own, the one that all dogs would eventually belong to if humans stopped messing with their gene pool.