Not really, not yet anyway, unless the government decides to become a duly certified registry, competing with all others out there.
Feral horses are a mixture of domesticated horses and when breeding same to same, you may get who knows what, their gene pool is immense, not limited by any one breed purpose.
That is antagonistic with breed purposes, that are breeding to keep distinct breed characteristics in the breed’s offspring.
Most feral horses are trying to get back to the original wild horse their basic genes are built on by domestication, the typical “brown dog” in the dog world.
If feral horses are left to reproduce at will, after not that many generations most will be scrubby 14 hands, is what being a wild horse works best in wild horse environments, perfect for that kind of species.
Our feral horse, from a herd in Nevada, had enough draft crossed that he was a hefty, big platter feet 15 hands, but would have been an outlier if such kept reproducing freely, with less and less of those making it thru the abundant grass and then famine stretches.
As he himself barely did, he had “rickets” in his knees, from growing up thru a drought as a yearling/coming two, as per our vet.
There is a lot more to feral horses, great as they are as individuals, to determine they are now a breed in themselves. Maybe in some more decades some may be.