Is the Game Engines Coodinate System different? I tried to rotate an object around its z-axis, by using a motion actuator with z-rotation (blue axis in screen) but it rotates around the y-axis (green). 

Are y and z-axis for some reasons swaped in the BGE?
No - Suzanne (the monkey) has her Z axis pointing out from her face, and the Motion actuator defaults to use local coordinates. Either:


Apply rotation to the mesh (Object>Apply>Rotation), or
Switch the actuator to world coordinates (press the L toggle button on the right of the Rot row).