In the Kronecker matrix product $C = A\otimes B$ we have that $C(i,j)=A(i,j)*B$ where the elements $A(i,j)$ are just numeric scalar values.
What if the $A(i,j)$ are matrix operators which act on $B$? Is there a name for that formalism or matrix product?
I don't think that this operation has a name. In fact, this operation can be conveniently with the help of the Kronecker product, e.g. for the $2\times 2$ case the product you give is $$ \begin{bmatrix} A_{11} B & A_{12}B\\ A_{21}B & A_{22}B \end{bmatrix} $$ which is the same as $$ \begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} B & 0\\ 0 & B \end{bmatrix} = \begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{bmatrix} (I_2\otimes B) $$ (where $I_2$ denotes the $2\times 2$ identity). The $n\times m$ case is similar, replace the $2$ by $m$.