Well jubily_, I can tell you why you are having problems. You do not have an official game. Based on those pictures you posted, I would say that that is definitely a pirated game. The circuit board is built differently than all Game boy cartridges I have seen. The outside cover of the cartridge also is obviously a poorly done cheap imitation.

At the top, it says "GAME" instead of "GAMEBOY Color". The cartridge is shaped slightly differently than the standard Game Boy cartridges, and the label looks like it is game art that someone printed out on a color printer. As you can see in the picture I linked to earlier, the cartridge should be clear. All Game Boy Color cartridges that are not backwards compatible to the original Game Boy are clear, while the gray cartridges, such as your game, were only used for pre-GBC games.

I would guess that someone tried to copy the game and make fake versions to sell, but they just didn't build the saving mechanics properly.