No, we're not forgetful; we simply never learn this woman's name. Neither the narrator, nor the hotel staff, nor even her own husband ever address the American wife by her name, which means that we can only talk about her using the words we're given by the story – American wife, American girl, and the title used by the maid and the padrone, "Signora." We'll admit—it's a little awkward since she's also the main character of the story, but there's a reason behind this awkwardness that's a bit deeper than you might expect. By not naming her, Hemingway helps us ask how much this character is an "American wife"—a stereotype or reflection of a typical "American wife" in the 1920s. Let's consider this, shall we?
The fact that the husband and wife are the only Americans in the hotel draws particular attention to their nationality. Hemingway is putting them in a context where they are the most American thing since Wonderbread. We can't be sure why they're here—maybe a honeymoon, a trip, or the husband's work. What do you think? Their drifting state also helps us see the importance of their marriage: being a wife is the only identity the woman seems to have. While her husband may have his reading (and may choose to listen or not listen, care or not care), the wife is simply waiting, as if she's dependent upon his movements, desires and decisions. Or, at least, she seems to be.