Leaving the artistic merits to one side, The Matrix is not Plato's allegory of the cave. Yes, they both share the element of release from captivity, and body-self dualism (which is crap, by the way), but half of the allegory of the cave is about metaphysical hyper-realism, the belief that forms are more real, more actual, than the things they're forms of. Those figures with the shadows projected on the wall, in the eponymous Cave? Yeah, that'd be the hyper-realistic forms; the multiple shadows are the multiple examples of each form, created by participation. If the Matrix were the Cave, it'd have a few separate people, AI or human, who each projected a number of different "selfs" into the Matrix. I'll grant it gets more Cave-like once the multiple Smiths show up in the second one, though.
Plato's concept of participation is one of the greatest achievements of all thought. There is a question in philosophy, "How can we talk about chairs, if there are multiple chairs and they're all different?" Plato's solution was that each chair you meet, partakes of the Form or essence of "chair-ness"; there is more than one of them because none of them participates perfectly, but only limitedly. This was genius—certainly better than materialism, which consists of saying that the shadows on the cave wall are the only thing that's real and the figures are a myth. Troglodytes.
But Plato took it too far, saying that the only real thing was the Form, what's participated in. There is no spoon, not because this is a simulation or something, but because this spoon is only a limited projection of the True Spoon—and there's only one of it.
Aside from the One and the Many, the other great question of Philosophy is Change/Constancy, and the related question of coming-to-be and passing away. Many Greek philosophers, including Parmenides of Elea and his student Zeno, hold that change is an illusion, because there's no reason that anything that exists should ever pass away, or come into being if it didn't exist. Conversely other philosophers—Heraclitus and the advaita school in Hindu thought—hold that, since there's no intrinsic nature to anything, everything is in a constant state of flux (which is the source of suffering for a Buddhist, remember).
Aristotle got around the question by, as he usually did, shooting for the Golden Mean. Obviously change is real—"Where are Parmenides and Zeno now?" as he so succinctly put it (they were dead). But obviously constancy exists, too—"Heraclitus has said that more than once, and yet used the same words each time." The reconciliation is the other great achievement of Greek philosophy: Act and Potency. In between the two irreconcilable states, Being and Non-being, and the shift between the two that was causing the trouble, he posited a third state: Being-in-potency. Anything that is not a contradiction in terms has the potential to exist. But only some things actually exist, when individual things participate in their Form (an idea he borrowed from Plato).
Unlike Plato, Aristotle's forms are not hyper-real; most of them are only real when there are individual samples of them. Hence, Mitigated Realism. For Aristotle, see, the form is one of four Causes, so called because each is the reason for something about any given object of thought.
- Formal: What a thing is.
- Final: What a thing is for (identical with Formal, if it wasn't made by people).
- Material: What a thing is like (its traits, including all its material properties).
- Efficient: How it got to be how it is, including how it came into being.