To be brief, it is a short short story and not a prose poem because it has at its center a character who yearns.
Fiction is a temporal art form. Poetry can choose to ignore the passage of time, for there is a clear sense of a poem being an object, composed densely of words, existing in space. This is true even when the length of the line is not an objectifying part of the form, as in a prose poem. And a poem need not overtly concern itself with a human subject. But when you have a human being centrally present in a literary work and you let the line length run on and you turn the page, you are, as they say in a long storytelling tradition, “upon a time.”