Difference between Excess property and property changes of mixing
The difference is subtle, however they are different as you suspect. 

You can look at the property of mixing as the change in property resulting from mixing the pure components (with everything at the same temperature and pressure) together to form the mixture.

The excess property is deviation between the actual property and the property in an ideal mixture at the same temperature and pressure. 

They are the same for volume and enthalpy because the volume of mixing and the enthalpy of mixing in an ideal solution are both 0. This is not the case for G or S.

In other words the difference arises because $M^{id}=\Sigma x_i M_i$ is not true for all properties. It is only true for enthalpy and volume. So the second term in your expression for the excess property above, cannot in general be written as $M^{id}=\Sigma x_i M_i$ (if it could, the excess property and mixture property would be the same). It should instead be $M^{id}=\Sigma x_i M_i + M_{mix}^{id}$. 

The extra term for mixing the components would be be given by $$
\frac{G^{id}_{mix}}{RT} = \Sigma x_iln(x_i)
$$
and 
$$
\frac{S^{id}_{mix}}{R} = -\Sigma x_iln(x_i)
$$
for Gibbs free energy and entropy, respectively.