Animal caretaking should be the skill used by dwarves to treat any injured pet assigned to them (as long as they have the labor enabled), as well as feeding any hungry/thirsty animals on chains (by default, only grazers).
Currently, however, the first part does nothing for animals (see below), but dwarves can gain experience in it during worldgen, leading to animal caretakers appearing in migrant waves. The second part, feeding chained grazers, grants no experience, but can satisfy "stay occupied" and "help somebody" needs. Note that dwarves will not feed animals in loose (stockpiled) cages, but will feed animals in built cages (if the food has seeds, the seeds will remain in the cage; they can only be removed by dumping).
Veterinary medicine is an expected feature that has not yet been implemented into the game, so animal caretaking does not actually serve its healthcare purpose as of yet.Bug:349 Wounded animals will instead heal on their own - or not at all, which makes even your greatest animal caretakers equivalent to peasants. Animal caretaking is not planned to be implemented in the near future, and as such, animal caretakers who arrive in migrant waves should be immediately repurposed into other, actually functional professions.
While non-functional in the vanilla game, the utility DFHack can be used to make animal caretaking work through the dwarfvet enable command; when the plugin is active, injured pets will rest in any activity zone marked as both a hospital and a training zone, where dwarves will then provide them with the necessary care. The command dwarfvet disable can be used to disable the plugin if necessary.