How can we say that the additivity of electric charges is much different from the simple addition of mass?

The quantisation of electric charge, q = ne, applies to electric charge only, even though charge cannot exist without mass.

No, there may be some charges inside the system. The overall charge is zero because the total positive charge will cancel out the total negative charge inside. This directly follows the principle of additivity of charge, where the positive and negative charges are the algebraic sum of each other. 

Even though charge cannot exist without mass, we can only have the quantisation of electric charge using the equation, q = ne. We look at the number of point charge and the electron's charge. 

One rule to think here is that electric charge is a scalar quantity. It has magnitude. It has both positive and negative signs, based on the direction it is forced to move in an electric field. Mass is always positive, on the other hand. So when you add mass, it never becomes zero. 

No, the system can still have charges inside. The overall charge is zero only because the total positive charge will cancel out the total negative charge inside. This directly follows additivity of charge rule. The positive and negative charges are the algebraic sum of each other.