Is the quantization of the harmonic oscillator unique? To put it a little better:

Is there more than one quantum system, which ends up in the classical harmonic oscillator in the classial limit?

I'm specifically, but not only, interested in an elaboration in terms of deformation quantization.
 A: For every classical Hamiltonian $H(p,q)$ there are infinitely many quantized systems that reduce to it in the classical limit. 
The reason is that adding to $H(p,q)$ an arbitrary number of expressions in $p$ and $q$ where at least one factor is a commutator doesn't change the classical syatem but changes the quantum version. To be specific, take, for example,
$H'=H(p,q)+A(p,q)^*[p,q]^2A(p,q)$ for an arbitrary expression $A(p,q)$.
This holds for systems in which $p$ and $q$ are canonically conjugate variables. It is easy to generalize the argument to arbitrary quantum systems.
In short:
Quantization of individual systems is an ill-defined process. 
Geometric quantization is more well-defined, as it effectively quantizes not a particular system but a particular group of symmetries. In the above case, it shows how to go from the classical $p$ and $q$ (i.e., from the classical Heisenberg Lie algebra with the Poisson bracket as Lie product) to the quantum version, but not how to go from a particular classical Hamiltonian (and hence a particular classical system) to its quantization.
A: 1) There are many inequivalent quantum systems that have the same classical limit $\hbar\to 0$. 
2) For instance, assume for simplicity that the quantum system is described by a single pair of creation and annilation operators, 
$$[\hat{a},\hat{a}^{\dagger}] ~=~\hbar {\bf 1},  \qquad\qquad 
\hat{a}~=~\sqrt{\frac{m\omega}{2}}(\hat{q}+\frac{\mathrm{i}\hat{p}}{m\omega}).  $$
The standard quantum harmonic oscillator Hamiltonian read 
$$\hat{H}~=~\omega (\hat{a}^{\dagger}\hat{a}+\frac{\hbar}{2}).$$
We can e.g. add any term of the form $\hbar P(\hat{a}^{\dagger}\hat{a},\hbar)$ (where $P$ is a polynomial) to the Hamiltonian $\hat{H}$ without affecting the classical limit.
In particular, we can shift the zero-point energy term.
A: This is not a complete answer to your question but quantization of the harmonic oscillator means quantization of the complex projective space $CP^n$ (as reduced phase space of the system).
In the context of deformation quantization you may then have a look at the paper A Remark on Nonequivalent Star Products via Reduction for CP^n by Stefan Waldmann, available on the Arxiv here, where inequivalent star products of $CP^n$ are discussed.
