Well, if it was flayed from someone/thing that was under her control, the enchantment would only need to handle movement. Possibly strip away the humanity from a rival vampire, so that all that remains is a mobile skin?
But it doesn't act or look like skin. Otherwise Harry would have been more adamant about how gross it looked. And it's described as looking like fabric, so it's not likely even leather.
I'm trying to think of any enchantment we've seen that's even remotely like what the fabric is. Short of spirit possession animating objects (like Bob and the statues), we haven't seen that level of manipulation, other than Before's tendrils, like TCF said.
But I took that as more that Before is used to having tentacles, and he made something that would let him control them like regular limbs. I don't think they would have worked if detached.
We've seen the Wamps mess with Little Folk before, using them as lamps. I'm wondering if they're doing the same with the cloth, imbuing it with some spirit creature similar to Bob but not in his class.
And depending on the level of cooperation, would Harry do the same with his duster to make it similar to Strange's cloak of levitation?