Suppose $\mathbf A$ is $n\times n$ and $\mathbf B$ is $m\times m$. Is it possible to write $$ (\mathbf I_n\otimes\mathbf I_m)+(\mathbf A\otimes \mathbf B)$$ as a Kronecker product between two matrices?
Obs.: The answer to this question is not exactly applicable, but seems to be related.
As explained in the answer to the question you cited, the equation: $$ C\otimes D = I_n\otimes I_m + A\otimes B $$ where $C$ and $D$ are unknowns, admits solutions iff $A=\lambda I_n$ and $B=\mu I_m$.