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This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two-dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.

This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two-dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.

Added

In response to Anirbit's comments above. Here's perhaps another answer which is more in the spirit of the original question.

Any filtered associative unital algebra $A$ whose associated graded algebra $\mathrm{Gr}A$ is commutative may be thought of as a quantisation of $\mathrm{Gr}A$ with the Poisson structure induced by the commutator in $A$. So an example of the situation you are after could be an associative unital algebra admitting two different filtrations with commutative associated graded algebras. I'd be surprised if one could not cook up something like this. It's another question altogether whether there are any "natural" examples.

This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two-dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.

This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two-dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.

This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.

This may not answer the question in the way you envisage it, but it is certainly possible for two different classical systems to give rise to equivalent quantum theories. One particularly well-understood example is the boson-fermion correspondence in two-dimensional conformal field theory as explained in Chapter 5 of Bombay lectures on highest weight representations of infinite dimensional Lie algebras by Kac and Raina.

There are also non-conformal examples, also in two-dimensional physics. One of the earliest known examples is the duality between the Sine Gordon and Thirring models, which are quantum mechanically equivalent yet very different classically.

In general, this is what S duality is about. This idea pervades much of modern theoretical physics, but its origins are in the Kramers Wannier duality in statistical mechanics. The Onsager solution of the Ising model is another example.