What implications are being made by the man being called "the man" and the woman being called "Jig?"
Hemingway's style of telling this story places the reader in the role of an eavesdropper. The reader only knows what he can see and overhear. This has sometimes been called the fly-on-the-wall perspective. It eliminates most exposition and gets right into the dramatic conflict. There would be no benefit in our knowing the man's name or the girl's name. We only know her nickname is Jig because the man calls her Jig. If Hemingway thought it was important for the man to have a name, he might have had the girl call him John or Bob--or whatever. Since there is only one man in the entire story, it is not necessary to have him identified by name. There are two women, so one is identified both as "the girl" and as "Jig," while the waitress is "the woman."
There is something very "modern" about this way of telling a story. We live in a world in which we see thousands of people who are complete strangers. We see many incidents which we don't understand and while will never be explained to us. We are strangers in a world of strangers. One word for this is "anomie." Another is "alienation." One of the reasons for this alienation is speed. We are speeding past a liquor store and see several police cars parked in front and some men being questioned and a small crowd of onlookers gathered. What happened? We will never know because we are already a mile past the scene and can only speculate.
Jig and the American seem especially alienated because they are foreigners in Spain. Jig hardly understands a word of Spanish. We don't know where they are going except that they are waiting for the train to Madrid. They probably don't know where they are going themselves. The American wants Jig to get an abortion, but he is going to have all kinds of troubles finding a reasonably competent person to perform the operation and keeping Jig persuaded to go through with it. There is the possibility of having trouble with the law. There is the possibility of something going wrong during the abortion procedure. The American will have to cope with Jig's inevitable feelings of depression, guilt, resentment, and regret, even if she doesn't have any adverse physical reactions to the abortion. And their relationship will never be the same.
I would make sure that one does not err too much on the side of reading into certain elements of the short story. Hemingway's work is so complex and so thick with implications regarding the characters in it and what can be taken away from it that it is important to be able to view as much analysis of it as more of a collection of thought and not too much weighing in one aspect or another. I do think that Hemingway might be suggesting that one of the challenges in all relationships between men and women is that the choices facing women are different, to an extent, than those of men. Jig is pregnant and while the American can point to different ways in dealing with it, in the end, she is going to have to bear the responsibility for it. In this light, giving her a name is representative of how she must own this and how her decisions will change her own life. "The American" is free, to an extent, to collect more luggage labels and move to more places. She has a name because she will have to own the condition in which she is. He is more "of the world" because, in the final analysis, he is able to walk away from this situation. In giving the woman a name and leaving the man to be more vague, Hemingway might be making a statement as to how women have to confront a different set of choices in the advancing stages of relationships than men do. The predicaments that women face are ones that, on some level, have to be owned by them, something that dawns on Jig at the end of the story. Her name is representative of how she, in the end, will have to make the situation her own.