The job of the sniper is to be "invisible" to the people he is targeting. The goal of the sniper is as much to terrify as it is to kill. You don't know where the shooter is hiding, so you don't know where to run, what to get behind for cover. It's almost as if some supernatural force is killing people. By giving the sniper a name, the author makes him a person. By keeping him anonymous, the author makes the sniper seem more monstrous and just another weapon of war.
In this story, two snipers are firing at each other. Each represents one side in the "war." Our sniper, the one through whose eyes the narrator tells the story, has to kill the old woman because she was going to reveal his position. His job is to be unseen. His job is to be that monster shooting from nowhere. At the end of the story, we discover just how much a monster he really is.
The most likely answer to this question is because the author's point is not exactly WHO the sniper is, but what he is doing and what his "job" entails and the people it affects, as well as how it affects him. Adding a name for the sniper would not add to the story, to be quite honest. Also, the author might not have given the sniper a name because of the commonality of them in Ireland during the civil war. The war was a bloody, violent one and it affected the entire country. Perhaps the author wanted the reader to be able to realize how this civil war affected everyone, including those who were on opposite sides of the civil war (within families).