Does partial charge violate the law of quantization of charge?
Both are.

Any isolated object has integral charge, as any elementary particle has integral charge. However, as we know, elementary particles are not truly located in one point. 

When we talk about stationary states in physics, we can assume, that particles inhabit some volume with special density function indicating how much of the particle in question is in any given unit of volume. When we consider isolated molecule, we can assume that it is in a stationary state. This means, that electrons may be viewed as electron clouds, and different parts of said cloud may be assigned to different atoms in the molecule. In case if total amount of electron density, assigned to specific atom is non-integral, we gain a partial charge.

Whenever atomic partial charges are estimated, there is a significant arbitrariness in the process, as there is no unquestionable way to define, which parts of electron density assigned to any given atom, as there is not strict definition of atom borders (atomic radii are estimated using equilibrium positions in some circumstances, but in truth electron density is non-zero even far beyond that radii). Still, the concept is useful enough.