There is already a good answer from ACuriousMind.

1. Concerning OP's title question: Yes, for functions $f,g\in C^{\infty}(M)$ of the classical phase space $M$ (which is a Poisson manifold), it is possible to replace the commutative, associative pointwise product $$\cdot~:~ C^{\infty}(M)\times C^{\infty}(M)~\longrightarrow ~C^{\infty}(M)$$ with a non-commutative, associative star product $$\star~:~ C^{\infty}(M)\times C^{\infty}(M)~\longrightarrow~ C^{\infty}(M),$$ e.g. the Groenewold-Moyal star product star product.

2. The idea is to turn the function-to-operator quantization map $$\Phi~:~ (C^{\infty}(M),+,\star)~\longrightarrow~ ({\cal A},+,\circ)$$ into an algebra isomorphism, cf. the topic of deformation quantization.

There is already a good answer from ACuriousMind.

1. Concerning OP's title question: Yes, for functions $f,g\in C^{\infty}(M)$ of the classical phase space $M$ (which is a Poisson manifold), it is possible to replace the commutative, associative pointwise product $$\cdot~:~ C^{\infty}(M)\times C^{\infty}(M)~\longrightarrow ~C^{\infty}(M)$$ with a non-commutative, associative star product $$\star~:~ C^{\infty}(M)\times C^{\infty}(M)~\longrightarrow~ C^{\infty}(M),$$ e.g. the Groenewold-Moyal star product star product.

2. The idea is to turn the function-to-operator quantization map $$\Phi~:~ (C^{\infty}(M),+,\star)~\longrightarrow~ ({\cal A},+,\circ)$$ into an algebra isomorphism, cf. the topic of deformation quantization.

3. By the way, in contrast the Koopman-von Neumann theory is an operator formulation of classical Hamiltonian mechanics where position and momentum operators commute.