Consider a Poisson algebra A (i.e. commutative algebra with Poisson bracket).

Let $\hat A$ be a deformation quantization of the algebra A. We know that construction of deformation quantization and more generally "formality isomorphism of Kontsevich" depends on choices of certain data (in Kontsevich approach we can change "propagator" and coordinates on manifold, in Tamarakin's approach we can choose arbitrary associator.)

Question Is it known/true/expected that for different choices of "that datum" we nevertheless obtain isomorphic quantum algebras $\hat A$?

May be one needs certain restrictions on setup (e.g. only smooth algebras A, only "generic" quantization morphism, whatever...) to guarantee uniqueness ?

Kontsevich also mentions that Grothendieck-Teichmuller group should act on the set of all deformation quantizations. Is it at least true that two quantizations living in the same orbit of that group give isomorphic quantum algebras ?

Question in formally precise form Consider a Poisson algebra. Choose two different formality isomorphisms (e.g. with different propagators or associators).

Define two star-products $\star$ and $\star'$ with the help of these two formality isomorphisms.

Question Are the algebras defined by these two star-products isomorphic ? More strongly - are these star products "equivalent" ? (See definition of equivalence in Stefan Waldmann's answer below or in Kontsevich paper).