Symbolism, Imagery, Allegory
Hemingway is an author who takes the material world very seriously. So before we get all crazy-analytical with things, stop and consider this for a second: that the cat in the rain is perhaps just a cringing, drippy, unhappy cat under a table.
Let's imagine this soaking wet cat for a minute. It has to be pretty pitiful. A horse or a dog in the rain is one thing, but there's something really wretched about a wet cat. It's basically trapped, too, beneath its little table-shelter in the plaza. The rain is pouring down so hard that it's trying to make itself as "tight" and small as possible to stay dry. As the American wife so perceptively says:
"It isn't any fun to be a poor kitty out in the rain" (30).
Hold up—did you catch that part where the cat was described as "tight"? A bit later in the story, Hemingway uses the same word to describe the way the wife feels around the padrone: "tight" Also with the silly quote above, we hear the wife sympathize with the cat. If we match this sympathetic statement alongside the shared adjective "tight," we'd say we have enough evidence to consider the cat as a symbol for the wife—or at least for some aspect of her.
There's something in the cat that the wife both wants and identifies with, which means that it's a symbol that works in at least two ways. Think about the cat's isolation, pitifulness, its lack of protection, and also the hostility of its surroundings.
All of these things remind us of the wife's own situation with her husband. She, too, is in an environment that's far from ideal, which explains why she might sympathize with the cat. The word that she uses in her statement about the kitty—"fun"—is also echoed later when she tells her husband that if she can't "have any fun," she should at least have a cat. When you're dealing with a writer who is as choosey with his adjectives as Hemingway, the repetition of a word is a big huge deal.
So, if the wife identifies with the cat's dire straits, what might she want from it? Well, think back to that adjective that they share: "tight." The cat's tightness and the "small, tight" feeling that the wife has before the padrone, are both instances of them being or feeling diminutive. It's the protection of the table that makes the cat tighten up and the largeness of the padrone that causes this in the wife. The protectiveness and respectfulness she sees in the padrone makes the wife want to tighten and draw towards him. She's experiencing her own sensitivity and vulnerability.
Amazing isn't it? The wet kitty doesn't get more than two sentences of face-time in the story, yet it's important to our understanding of the wife in two completely different ways. And never will you feel the same about wet cats again …