Is it possible to drive a 4x4 RGB common-cathode LED matrix with a MAX7219?
The answer is no. If the matrix has 4 cathodes and 4*3=12 anodes, and the driver has 8 current-sourcing pins, it is not possible. If the LEDs' pins were individually accessible, you could rearrange them to 8 cathodes and 6 anodes which would work, like this:



But still, there would be a problem with different perceived intensity for different colours.

Charlieplexing is not applicable here because direction of current is not reversible. 8 pins can source and 8 pins can sink