I cannot pretend to understand the question (I would answer what I understand as a clumsier version of @Qmechanic 's answer), but, as a courtesy of the casual reader, I'd rewrite the first equation into the equivalent format of the second one, so as to make it easier to compare and contrast apples with apples.
The classic Groenewold star product $$ (f \star g) (x,p) = f(x,p)~ e^{\frac{i \hbar}{2} ( \overleftarrow{\partial_x} \cdot \overrightarrow{\partial_p} - \overleftarrow{\partial_p} \cdot \overrightarrow{\partial_x})} g(x,p)=\left . e^{\frac{i \hbar}{2} ( \partial_x \cdot \partial_{p'} - \partial_p \cdot \partial_{x'})} f(x,p)g(x',p') \right \vert_{x'=x,~ p'=p} $$ is routinely recast as $$ (f \star g) (X) =\left . e^{\frac{i\hbar}{2} \epsilon^{\mu\nu} \frac{\partial}{\partial X^\mu} \circ\frac{\partial} {\partial Y^\nu}} f(X) g(Y) \right \vert_{Y \to X}\\ X^\mu\equiv (x,p) ~. $$ The $\circ$ symbol indicates multiplying/dotting the components of the first symplectic vector crosswise, i.e. coordinates multiply momenta and vice versa. It should be apparent how to generalize this to multidimensional phase spaces, with r space coordinates, wrinkle their geometry, and append degenerate dimensions, as the proper answer details. You may be suggesting this with the dots dotting r x components to r p components, as with the symbol used, in which the expression is already generalized to higher r.
In the mathematics literature, the * product has been extended to recondite spaces that more than cover any NCG ever contemplated.