A metaphor is a comparison that does not use the words like or as. A dead metaphor is one that has become completely divorced from the imagery that inspired it. For example, "time is running out" once conjured the image of the last sands running out of a hourglass in the days before clocks were common. Now, although we know what the words mean, they have lost the connection with their original imagery. A live metaphor, in contrast, still evokes an image in a vivid way. For example, "my love is a red rose" is a live metaphor because it conjures the beauty of a rose in our minds.
In this simple, straightforward story, the literalism of the telling means there are few metaphors. However, the wooden bowl from which the old man has to eat is a metaphor for how his son and daughter-in-law have come to devalue him for aging. This is a fresh metaphor, though old fashioned, because it brings to mind the image of a rough wooden bowl. It becomes even more poignant as the grandson starts to fashion such a bowl out of bits of wood for his parents, showing through the bowl metaphor that the boy is learning the lesson of devaluing the elderly.
In terms of structuralism, the story is a fable: a short tale, usually a folk tale, with a moral lesson at the end. The meaning of the story emerges from that structure. The characters, for example, are generic, universal symbols of three different generations—old, adult, and young—without given names.
Finally, the sequence of events adds to the power of the story. The son and daughter-in-law are first shown as hard-hearted, and we are prone to be angry at them for their treatment of their father. However, we see them learn the lesson of empathy as they are able, through their son's words and actions, to see themselves in the grandfather's shoes. The story is effective because of the reversal: the parents learn the compassionate lesson of do unto others as you would like others to do unto you through their ability to reverse their point of view. They move from seeing the world through their own eyes to the eyes of another.