There were no horses in North America until the Europeans arrived. Therefore, it makes sense that the overwhelming majority of horses the Indians had access to depend on the Europeans who settled/conquered that particular region. Spanish horses were likely prevalent across the south and west of the continent, English on the East coast, French in the midwest, etc. Some of the horses could certainly be considered mustangs, as some were escaped horses. Many more were acquired in trade or taken in battle, though.
I think it takes a lot of time and foundation stock to truly develop a breed, and most Indian tribes simply didn't have enough of either of those to really do that. If anything, I'd say they are probably more like selectively bred mustangs (i.e. "mutts" of various European strains) than anything else, ultimately coming from the same stock, even if some never were feral.
As any breeder can tell you, some traits are easier to selectively breed for than others. Color, in particular, can make it seem like a breed is/was established, but a breed needs to also have consistency in conformation, temperament, and a wide enough gene pool to avoid inbreeding.