They think they are a breed...but they are not. Something being a breed doesn't hinge on having an AKC or UKC recognition. A breed is a population of animals that breeds true to type.
The Am Bully currently has 4 classifications of dogs: Classic, Standard, XL, and Extreme.
Classic is very similiar to the American Stafforshire Terrier, just a bit heavier built. Standard are heavier yet, wider, and with bigger heads. XLs increase the weight and width even more, even bigger heads, and are taller as well, but usually gaining more weight than height in comparison to the 'standard' dogs. Extreme is wider than the standard, with bigger heads, but shorter.
The Am Bully group keeps on adding in new catagories so no matter what is produced it is a champion of something. This is why the Am Bully isn't a breed, because they can't manage to decide on a type then selectively breed the foundation until it is producing true to type offspring.
As far as what has gone into this 'group' of dogs? The answer is anything. Right now you can pick up a pitbull looking dog from the pound and get it registered as an Am Bully. Your pound find may be pure pitbull or it could be a boxer x lab mix, you can still register it as an Am Bully.
MOST of the Am Bullies, from the start, were pitbull mixes, ABPTs, and Staffs. People were selecting for the widest heaviest ones. To go even further Bulldog was bred in for fat heads. Mastiff, cane corso, and other big dogs contributed for size.
'The bigger the better' is currently rampant in the American dog world. I am seeing Labradors that are oversized, and not just because the dogs are putting on more pounds due to lack of exercise. When I see people asking for German Shepherd, they are always asking for the biggest ones, usually they want 100 pound miniumum or brag about having 110 or 115 pound animals. The truth is dogs give up agility and athleticism as they get bigger. These giant labs cannot jump from the boat as cleanly, swim as fast, or work the fields for as many hours. GSD at or over 100lbs see a marked slowdown in their sprint speeds, jump heights, and similar metrics. 80 and 90 pound dogs can do a much better job. Even for bear hunting, people aren't using Great Danes, they are using 60 pound hounds who are long legged, fast, spry, have great stamina, and who can dart in and out biting the bear while dodging the claws and teeth.
For the Am Bully, it's even worse. The bulldog infusuion and desire to have a wide fat face has created a dog that can't breath as well as an ABPT, especially not once they have a full bite on. The desire for width has caused the legs to be placed further out on the shoulder which just mechanically is a disadvantage and greatly reduces run speed and jump height. This is in addition to basically the same skeletal structure carrying more weight.
FURTHER, the true 'pit' which is best encompassed in the ABPT has been bred to fight other dogs as entertainment and gambling. APBTs are very dog aggressive. But the flip side of this is pits were actively bred to pretty much like everyone because the breeders of dogs would sell adult dogs and trade adult dogs all the time. A dog that was attached to the old owner and attack the new owner = BAD.
The 'pit' in ghetto culture was expected to be an attack dog and guard dog, which was entirely contrary to it's actual breeding, so dogs that had the BAD trait of disliking people were actually seen as better, plus there were purposeful breedings with dogs like rottweilers, pit mixes, and mutts to make better guard dogs.
THIS subset of pits, staffies, and mixed pits are what formed the breed. So you end up with a real temperment mess. Some Ambullies are being bred to be aggressive to be guard dogs others are being bred to be docile pets. Furhter because of all the conflicting mix of temperaments that got into the earliest dogs, the Am Bully can be a real temperament mess.
As for the main purpose, the dog is bred to 'look tough' (even though it is much less capable than an actual ABPT of accomplishing any sort of physical task) which isn't any better or worse than why we have the British Bulldog, toy poodle, dachshund, afghan hound or all sorts of other breeds that once long ago may have been created for a real purpose but have since been morphed to their current form simply because we humans find it eye pleasing.
Closing Note - few people realize exactly what the real ABPT looks like and is. The APBT rarely has cropped ears like the other breeds often are given to make them look more intimidating. The APBT male is usually 50 pounds give or take 5 pounds. They are thin and wirey, muscles everywhere on a small tight frame, with a small powerful chest. The ABPT is the 'Bruce Lee' of the dog world, not the Arnold Schwarzenegger.