Why would author Tim O'Brien continually use the full name and rank of the soldier in the story (Private First Class Paul Berlin) instead of referring to him as Private, or Paul, or Berlin?

This is Tim O'Brien's way of emphasizing how, in the middle of a war, a soldier loses his identity; how he becomes a member of a platoon rather than an individual. Back home, Paul Berlin may be Mr. Berlin, or just plain Paul to his friends and family. But out in the jungles of Vietnam its a different story altogether. There, he's Private First Class Paul Berlin, part of a much bigger whole: a fighting unit in which everyone pulls together to protect and defend each other.
One of the themes of "Where Have You Gone, Charming Billy?" is the conflict between the individual and the collective, and Paul's full rank illustrates the ever-present tension between Paul Berlin the man and Paul Berlin the solider. Paul is only in Vietnam because he was drafted; he didn't choose to be there. But if he's to overcome his deepest fears, he needs to put his former identity as a civilian behind him and think and act like a soldier instead.