No, mustangs are not a breed. Mustang means feral; any horse that is captured out in the wild would be considered a mustang.
The word mustang comes from the Spanish word, Mesteno. While there are registries, the horses come in many sizes and shapes, plus color patterns. You can register them as appaloosa, pinto, quarter pony, American Warmblood and just about any other 'breed' that doesn't require parents to be known.
As far as what to call the domestic bred foals, yes, people still do call them mustangs. Most will specify that they are domestic bred. My opinion on the matter is that they need not be reproduced in captivity as there are so many being held in corrals that need homes. There may be exceptions, as people are heavily promoting the Kiger, or some horses carry some sort of color that other folks may take an interest in. Most are just nice range horses, good trail horses, wonderful children's mounts. They're used in reining, dressage and for police work and by the US border patrol. But most shouldn't be adopted for the specific purpose of breeding.