quantum mechanics - Inequivalent quantizations of a classical system

Suppose to have a Hamiltonian $H(q,p)$ defined on a phase space with (q,p) and suppose that exist a canonical transformation

$$(Q,P)=X(q,p)$$ such that the classical dynamics is equivalent using $H'(Q,P)$ and (Q,P). Now, what happens if I quantize the system using (q,p) or (Q,P)? I would say that I get two Hilbert spaces E and E' that are not isomorphic in general because a canonical transformations could be non linear. Is it true? If so, why we are used to quantize a system in a specific (q,p) phase space? I observed this ambiguity in the quantization of the hydrogen atom: the system is quantized in the phase space $(x,y,z,p_{x},p_{y},p_{z})$ and only in the quantum world we shift to polar coordinates: what would happen if this change is made at a classical level and then we quantize the phase space (r,$\theta$,$\phi$,p$_{r}$,p$_{\theta}$,p$_{\phi}$)?