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.