You CAN synchronize a GENUINE controller with the Wii. AFAIK, you can only use it DIRECTLY in certain homebrew applications such as Nintendont and some emulators or loaders. You can't use it in the Wii Menu or Wii/WiiWare/VC games, except through a third-party ADAPTER that connects to the Wii Remote and allows emulating a Wii Classic Controller.
With counterfeit ones, I suppose something similar happens, although I've heard that some of the aforementioned homebrew applications and adapter may not recognize them.