Both Plato's "Allegory of the Cave" and Orwell's "Shooting an Elephant" involve themes of oppression. In the "Allegory of the Cave," Plato suggests that people have a false sense of reality. He expresses this with the allegory that people are inside a cave, chained in such a way that they can only face one way, and that is toward the inner wall of the cave. People only see the shadows on the wall which are projected by the figures and fire behind them. People are even further removed from the ultimate Truth of reality which is outside the cave. The general meaning here is that until we think deeply and/or break the bonds of our mental/physical enslavement, we will always see the world in this false reality and will therefore behave accordingly.
In "Shooting an Elephant," there are similar examples of people being conditioned to think and behave in certain ways. The narrator is a British officer stationed in Burma. He plays his role of representative of the colonizers (British) and the Burmese are relegated to their roles as the colonized. Because of this political structure of colonizer/colonized, each corresponding group is forced/conditioned to think and behave accordingly. Even though the narrator is in a superior role, he too feels forced to behave in certain ways (just as Plato's chained people are forced to think and behave by their limited perceptions of the world).
The narrator directly notes that despite feeling like a "leading actor" in his authoritative role, he too feels like he is a puppet: a clearly similar notion of Plato's chained cave dwellers, being manipulated by those chains as a puppet is manipulated by the strings.
Here was I, the white man with his gun, standing in front of the unarmed native crowd--seemingly the leading actor of the piece; but in reality I was only an absurd puppet pushed to and fro by the will of those yellow faces behind.
Note that this image of being "pushed to and fro by the will of those yellow faces behind" is strikingly similar to Plato's chained cave dwellers being manipulated by the figures and fire "behind" them. The narrator also acknowledges that the Burmese are unfairly oppressed and secretly is on their side. Thus, if he believed he could remove himself from his role as colonizer, he might take the Burmese side and help to liberate them from their chains.
Also note the enslavement of the elephant who is usually chained up and therefore trained to behave. It is only when he breaks his chains that he is able to experience some sort of freedom. His break does result in carnage, but the narrator does note that the elephant, in his freedom, is relatively peaceful when not provoked. "They all said the same thing: he took no notice of you if you left him alone, but he might charge if you went too close to him." This might be stretching an interpretation of what the elephant as thinking, but in connection with the oppression suffered by Plato's cave dwellers, the oppressed Burmese, and the narrator's own conditioned behavior, the elephant is simply trying to maintain his freedom. This is one of the important aspects of Plato's allegory: once a person becomes free of his/her chains, he/she will not want to return to that oppressed/chained state.