What is/are suggested by the names the two characters are given in "Hills Like White Elephants"?
Writer first introduces the male "The American" and the female "girl." He calls the American "the man" later and calls the female "girl" throughout the story. The man addresses the girl "Jig" but she never addresses him by anything but "you." And also, at the beginning the narrator introduces the two as "The American and the girl with him."
The man in the story is referred to as "the man" or "the American" his companion is referred to as "the girl". The idea that the man is a "man" while his companion is a "girl" suggests that she is young and innocent. Since they are discussing such a heavy topic, this suggestion may make the reader think that she does not have the authority to have an opinion about it. It is likely that this is the way "the man" views her.
The fact that she does not refer to him in any terms other than "you" while he refers to her using a name, suggests a disconnect between the two of them. Their relationship is unclear and confused to both them and the reader... the way they address each other indicates this confusion, as well as the phrase "The American and the girl with him" at the beginning of the story. Since it does not say "his girlfriend" "his wife" etc, we do not know if they have made any type of commitment to each other.