I assume you are referring to the short story entitled "The Man I Killed" in this powerful collection of stories. Whilst there is a dead Vietnamese man in the story "The Things They Carried," no mention is made of how it reminds the narrator of his first date. However, in "The Man I Killed," in which the author has to come to terms with his first kill during wartime, the date that is referred to is not the author's first date, but the first date of the Vietnamese soldier whose life the author reconstructs as he looks upon his body and tries to grapple with his grief and guilt at having killed him. Tim O'Brien constructs a life and a background for this soldier that he had killed, which of course makes his feelings worse:
And as he waited, in his final year at the university, he fell in love with a classmate, a girl of seventeen, who one day told him that his wrists were like the wrists of a child, so small and delicate, and who admired his narrow waist and the cowlick that rose up like a hird's tail at the back of his head. She liked his quiet manner; she laughed ta his freckles and bony legs.
Tim O'Brien seems to do this to humanise the war and what was to him a profoundly dehumanising experience. Giving his first kill a history somehow seems to go against the dominating "us and them" mentality that is so powerfully oppressive and forces you to see the enemy as just bodies to be killed. The background of the Vietnamese soldier reminds both us and O'Brien that he was human too.