I think this is a difficult question to answer, mostly because it's not clear whether we are attempting to understand the author's mind in making the choice not to name the sniper, or if we are analyzing the relevance and impact of this choice upon the story itself. I think it's always dangerous to presume that we can understand the author's choices by the finished work, and it is always better to say "what evidence supports", or, "why might the author have chosen", rather than to state our insights as conclusive.
One possible reason that the sniper is not given a name is that we are meant to relate to him; to envision that he could be ourselves, in the same situation. It may more specifically suggest that the sniper could have been anyone engaged in the war, to drive home the point that the possibility of fighting against or killing a family member was more common than not.
We might also question what the story would gain by giving the character a name; is there any specific name which would add to the point and purpose of the story, and might this compensate for losing the "everyman" quality that the use of no name provides?