Unlike any other species in the base game, bats cannot be hunted or skinned in Red Dead Redemption. They also do not randomly spawn in the gameworld. A bat swarm spawns around 6:00 P.M. at Tumbleweed Manor. They swarm over the mansion every evening. They also swarm out of an upper window in a building in Thieves' Landing. In this sense the bats are not truly creatures, just scripted atmospheric events.
In the Undead Nightmare DLC pack, Undead Bats feature prominently, and are the only flying creatures found while roaming the frontier (except vultures). Shooting and running through the brush often disturbs flocks of undead bats from their perch, and they will fly away in all directions. Although lock on does not work on them, they can still be killed with a precise shot or by using deadeye. It is also easier to shoot a bat if you're ontop of a ledge. The corpses can then be searched to obtain undead bat wings but they cannot be sold as money does not feature in the post-apocalyptic zombie world. The bats are rather large, this is evident by their wingspan when killed, seemingly over 4.5-5ft. This would make them too large to be a vampire bat, and closer to a megabat (Flying Fox,) although they may just be a nondescript bat species designed to fit the horror theme of the game.
The Tumbleweed Bats were most likely put in by Rockstar to add to the 'haunted' myths/themes in the newspapers and the other strange occurrences of the ghost town, like the dark horse and ghost people.
Behind the Scenes
Originally, the player would be able to hunt bats, as seen in this picture. It is odd that Rockstar would remove this item after it had successfully been implemented. A possible theory is because a bullet would realistically render the bat's body useless, similar to how the Buffalo Rifle disintegrates any small animal (i.e. rabbits, skunks) in-game. It could also be because a shotgun could easily take out an entire swarm of bats in a handful of shots, which would be excessively easy money. Another theory is that they were removed simply to be re-added in the Undead Nightmare DLC pack, as they were more fitting to the theme of the game.
As bats cannot be hunted by the player in the base game, they are not part of the "Unnatural Selection" achievement/trophy.