Any time you get a guy with wings running around in your story, it's a pretty good bet that the supernatural is one of the themes. In "A Very Old Man with Enormous Wings," we get dueling views on just what the winged old man means in terms of the creepy, spiritual, and otherwise supernatural. A neighbor lady, and many of the crowds, believes that the old man is an angel. The town priest, however, wants to do further testing. The point? The story gives us a look at everyday people's reaction to their contact with the unexplained. It's like Unsolved Mysteries, only without Robert Stack or the creepy soundtrack.
In "A Very Old Man with Enormous Wings," the supernatural stands in for anything that is different, to show how we treat outsiders.
The story shows the way that both rational and irrational responses to the supernatural are laughable, and points out how gullible people can be. (By the way, did you know that the word "gullible" isn't in the dictionary?)