Oh, certainly it is. Authors don't do things by accident - they have intentions behind each choice they make. Keep in mind however, that the girl does have a name of sorts. She is Jig. He, however, is only known by the fact that he is an American. He says at one point:
“It’s really an awfully simple operation, Jig. . . . It’s not really an operation at all.”
That could be her actually name or a nickname; that fact is never clarified. And of course, Hemingway doesn't refer to her by a name.
The symbolism here is that this couple could represent any couple in a similar situation. We are meant to focus on the conflict they are having and the ways they deal with it. Their dialogue shows us the different, and one could say, gender-specific reactions to the unwanted pregnancy. It is much easier for the man (also known as "the American") to disregard it. The girl obviously does not want to.
The man tries many means to persuade her not only to have the abortion but to be comfortable with the decision. He wants her reaction to be like his. She, however, wants to make him happy. She says that she'll have it if that what he wants, though it is clear that it isn't what she wants.
Hemingway gives readers a simple but in-depth portrayal of the differences between men and women, made more universal by the elimination of names.