This seems like a clasic analog issue. Does adding 1 spin to a 300 spin system add exactly 2^300 bits of information and 1 spin to a 2 spin situation exactly 2^2 = 4 bits of information?
Perhaps this will help clarify why it is different. In the classical case, adding one more spin is doubling the size of your set. In the quantum mechanical case, adding one more spin instead doubles the dimensionality of the space of possible states.