It's been about 5 years since I read Plato's Republic and, ironically, I'm reading some Plato at the moment which led me the think of the same correlation after I saw the second movie.
I think, however, that the philosophy behind Matrix is a bit more complex than that. Now, I haven't read, for example, Simulacra and Simulations, but it's my guess that, being that, from what I understand, it works on the same themes, it, and other philosophical writings like it, are more complex elaborations on the Allegory of the Cave.
Anyway, to apply it to Matrix, I'd say the obvious point that Neo is in the role of the philosopher, taking people out of the cave.
****SPOILER!!!!!!*****
However, if the double Matrix theory is true, then you run into a situation where they are in a cave within a cave. Shadows in the second cave bening caused by artificial lights in the first, and they have yet for a person to discover this in order to show the truth to to thers and set them free.
I envy my nieces for when they grow up and get to watch the movie in philosophy classes
