Can you find a schematic for the PCB behind the matrix?
I see 4x 74hc595 chips. I don't see any bypass capacitor or current limiting resistors. I think this is probably a poor design and would not expect it to have a long life. But it could be fun while it lasts!
It should be possible to mix 7 colours quite easily (red, blue, green, yellow, cyan, magenta, white).
If you want more colours than that, it should be possible, but the code would be beyond beginner level.
So I suggest we help you get those 7 colours, and if you decide you want more and are ready to learn some more advanced coding, the forum will give you all the help you want.