There is no good definition of what quantization is. It usually means that we start with a commutative structure $A$, a 1st order (non commutative) deformation $A\otimes k[h]/(h^2)$ of this structure and we want to lift it to an actual (or formal) deformation $A_h$ so that $A_h/h^2A_h = A\otimes_k k[h]/(h^2)$.
In some cases we know that the lifting exists but it is never unique. Instead, in the formal setting, there is a group that acts simply transitively on the set of quantizations. So the set of "quantizations" is a torsor under this group.
For example, quantum groups theory is a theory of quantization of semi-simple groups or more precisely of their Hopf algebra. Drinfeld showed that, given the enveloping algebra $A = Ug$ of a Lie algebra $g$ (it is co-commutative Hopf algebra) over a field of caracteristic 0, and an ad-invariant symetric tensor $t\in Sym^2(g)^g$ (corresponding to a first order deformation), there exists a quantization of $Ug$, namely a quasi-triangular quasi-Hopf algebra $A_h \simeq Ug[[h]]$ reducing to these data mod $h^2$. The set of (universal) quantizations is a pro-algebraic variety $Assoc$ of "associators" and it is a torsor under the Grothendieck-Teichmuller group $GT$ which is an extension of $G_m$ (because we have an action $\varphi:h\mapsto \lambda h$) by a pro-unipotent group (because we can filter it by $\varphi \equiv id \mod h^n$).
The theorem of Kontsevich on deformation quantization of Poisson varieties shows a similar pattern. Here the structure is associative algebra (with some additional properties). A 1st order deformation of a commutative algebra $A$ is a Poisson bracket. And we try to lift it to a formal deformation $A_h$.
In both cases, I think the choice of an associator gives you a functor. But a) we only consider "universal" quantizations. If we consider a single object (Hopf algebra or Poisson algebra) it may have some deformations that are not given by the universal recipe. b) in both cases, we just ask for an isorphism $A_h/h^2A_h = A\otimes k[h]/(h^2)$. We do not ask for a section $a\mapsto \hat a_h$ (a quantization rule like symetric ordering). I think such a requirement would always break functoriality.