Both the speaker and the dead man enlisted in opposite armies, but there's no difference between the two… well, except that one is alive and the other is pushing up daisies. That aside, the idea here is that the two positions these men are in are interchangeable in the speaker's version of events. Neither man has a name. They probably joined because they were broke. And they were both infantrymen in the same war. But we should also keep in mind that the speaker has made up everything he says about the other guy. He wants to give this guy an identity, maybe as a way to cope with the fact that he took the man's real identity away forever.
Line 1: We know nothing about the two men, and the speaker doesn't make much of an effort to tell us anything. And that's kind of the point. There's no devil in the details here.
Line 6: Even though the two men stare at each other face to face, they have no contact. To one another, each person is simply a face—an object to be eliminated. Chilling, don't you think?
Line 7: They both shoot at each other at the same time, so they are both equally guilty at this moment—it's yet another way in which these two dudes are alike.
Lines 10-12: Like in line 6, the speaker can only identify the other man as his foe or enemy. But that's a projection. Someone told him this other man was his enemy, and the speaker chose to believe it. No one's asking why.
Lines 13-15: The speaker makes up a story to give the other man an identity, which is yet another projection. And of course, it sounds like the story the speaker gives this man is an awful lot like his own story. That makes us wonder why the speaker wouldn't try to distance himself more from the man he killed. Why not demonize him, instead of view him as a friend?