Let $A$ be an $n \times m$ matrix, and let $\circ$ be the Hadamard product. What are sufficient conditions on $A$ for the following to be true for all $m$-vectors $x$ and $y$? $$ Ax \, \circ \, Ay = A(x \, \circ \, y) $$
I think one sufficient condition is the following: each row of $A$ is a standard basis vector (i.e., there exist $k_1, \dots, k_n \in \{1, \dots, m \}$ such that the $i$th row of $A$ is $e_{k_i}^T$). Then the $i$th element of the left-hand side simplifies to $$ \sum_{j=1}^m \sum_{k=1}^m A_{ij} A_{ik} x_j y_k = x_{k_i} y_{k_i} $$ and the $i$th element of the right-hand side simplifies to the same thing: $$ \sum_{j=1}^m A_{ij} x_j y_j = x_{k_i} y_{k_i} $$ Are there weaker conditions that also get this result?
Assume that $A\circ A=A,\,$ so that every element $A_{ij}$ must equal either zero or one.
Further assume that the sum of each row satisfies $\,\,\,\sum_jA_{ij} \le 1$
Such a matrix will satisfy your main equation $\,A(x\circ y) = Ax\circ Ay$
This is almost what you had, but it also allows $A$ to contain rows of zeros.
Exchanging matrices and vectors, a related problem is to find
those vectors which satisfy $a^T(X\circ Y) = a^TX\circ a^TY$
Using similar assumptions: $\,\{a\circ a=a,\,\,\sum_ia_i\le 1\}$
the only possible vectors are those of the standard basis and the zero vector.