Skin and hair cells are different. Skin cells are dead tissue, hair is mostly protein with some dead cells. Hide is an animals' skin, albeit covered with furs, or hairs of some sort.
For the purpose of DF, the hair is sheared off in the butchering process, (you can turn it into thread) and the hide left as unprocessed leather to rot in the sun. When all is said and down, technically the pieces are separated out.
Well, kind of. Skin cells are alive in the dermis before they become fully keratinized over the course of their life cycles, while hair cells, once fully keratinized, likewise die. That said, I think it's a bit too much to say that they're one and the same, any more than nails and skin are one and the same. All three include keratinocytes which produce keratin (in what's ultimately a self-destructive cycle as noted earlier), and populations of cells can migrate between the three within the integumentary system as a whole. However, once you reach the point where you're looking at hairs and skin as tissues or organs respectively, you should also consider the differing structures of the two, regardless of whether the two both have keratinocytes and melanocytes.
To bring it back to the matter of undead in DF, specifically, though, the two are indeed separated during butchering as you say, and heavens know how either of them move once reanimated. Magic, I suppose.