We establish conditions to characterize probability measures by their L^p-quantization error functions in both R^d and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the L^p-Wasserstein distance). We first propose a criterion on the quantization level N, valid for any norm on Rd and any order p based on a geometrical approach involving the Voronoi diagram. Then, we prove that in the L^2-case on a (separable) Hilbert space, the condition on the level N can be reduced to N = 2, which is optimal. More quantization based characterization cases in dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found at the end of this paper.