Can adding a copper plane to the 2.4GHz antenna amplify signal strength?
Can that copper plane act as a 2.4GHz antenna?


You bet. It's called a patch antenna.


  Also, can this plane "really" amplify something?


Technically, no, but in common usage yes. It's more accurate to say that a properly-designed antenna will be more efficient, and produce a higher output signal, than a poorly-designed one.

If you start out with an antenna which is just a random length of wire, the odds are very good that this will make a crappy antenna. Put a properly-designed patch on the end of the wire, and its output will increase. Did the patch "really" amplify the signal? If you don't understand how the antenna works, the answer will seem to be yes, but if you have more knowledge you'll say no, it's just a better antenna.