Kontsevich's formality theorem implies in particular that star-products on a $C^\infty$-manifold $M$, $$f\star g = fg + \sum_{k\geq1} \hbar^k B_k(f,g),\qquad f,g\in C^\infty(M),$$ where $B_k$ are bidifferential operators of degree at most $k$, are classified, up to the gauge equivalence $$\star \sim \star' \iff \exists\ T=I+\sum_{l\geq1}\hbar^lT_l\ ,\quad T(f\star g)=T(f)\star' T(g)$$ where $T_l$ are differential operators of order at most $l$, by Poisson bivectors depending formally on $\hbar$ $$\Pi(\hbar) = \Pi_0+\sum_{k\geq0}\hbar^{k}\Pi_k \in C^\infty(M,\wedge^2 TM)[[\hbar]],\qquad [\Pi(\hbar),\Pi(\hbar)]_{\mathrm{SN}}=0$$ (where $[\cdot,\cdot]_{\mathrm{SN}}$ is the Schouten-Nijenhuis bracket) up to formal paths in the groups of diffeomorphisms of $M$ starting at the identity diffeomorphism.

The star product commutator $[f,g] := \frac{1}{\hbar} (f\star g - g\star f) = \{f,g\}+\sum_{k\geq0}\hbar^kC_k(f,g)$ starts with the Poisson bracket associated to the Poisson tensor $\Pi_0$. So a star product, which is an associative deformation $(C^\infty(M)[[\hbar]],\star)$ of the associative commutative algebra $(C^\infty(M),\cdot)$, induces in particular a Lie algebra deformation $(C^\infty(M)[[\hbar]],[\cdot,\cdot])$ of the Poisson algebra $(C^\infty(M),\{\cdot,\cdot\})$.

1) Is there a sensible notion of "deformation quantization" of the Poisson algebra $(C^\infty(M),\{\cdot,\cdot\})$ as a Lie algebra deformation $(C^\infty(M)[[\hbar]],[\cdot,\cdot])$ which does not require referring to (or the existence of) a star-product, i.e. a notion of quantum commutator without the corresponding star-product?

If yes,

2a) does any such special Lie algebra deformation come from a star-product anyway?

2b) is there a classification analogous to Kontsevich's one?

Motivation: in field theory one often faces the problem that, while commutators of local functionals can be defined as local functionals themselves, star-products (or even just classical products for that matter) of local functionals are not local functionals (they are sometimes defined only in some completion of the tensor algebra of local functionals). Can one do without star-products and consider the classification problem for commutators instead?