Let the number of stones in row $i$ be $r_i$ and let the number of stones in column $i$ be $c_i$ . Since there are $m$ stones, we must have $\sum_{i=1}^n r_i=\sum_{i=1}^n c_i=m$
Lemma 1: If any $2$ pilings are equivalent, then $r_i$ and $c_i$ are the same in both pilings $\forall i$ .
Proof: We suppose the contrary. Note that $r_i$ and $c_i$ remain invariant after each move, therefore, if any of the $r_i$ or $c_i$ are different, they will remain different.
Lemma 2: Any $2$ pilings with the same $r_i$ and $c_i$ $\forall i$ are equivalent.
Proof: Suppose piling 1 and piling 2 not the same piling. Call a stone in piling 1 wrong if the stone occupies a position such that there are more stones in that position in piling 1 than piling 2. Similarly define a wrong stone in piling 2. Let a wrong stone be at $(a, b)$ in piling 1. Since $c_b$ is the same for both pilings, we must have a wrong stone in piling 2 at column b, say at $(c, b)$ , such that $c\not = a$ . Similarly, we must have a wrong stone in piling 1 at row c, say at $(c, d)$ where $d \not = b$ . Clearly, making the move $(a,b);(c,d) \implies (c,b);(a,d)$ in piling 1 decreases the number of wrong stones in piling 1. Therefore, the number of wrong stones in piling 1 must eventually be $0$ after a sequence of moves, so piling 1 and piling 2 are equivalent.
Lemma 3: Given the sequences $g_i$ and $h_i$ such that $\sum_{i=1}^n g_i=\sum_{i=1}^n h_i=m$ and $g_i, h_i\geq 0 \forall i$ , there is always a piling that satisfies $r_i=g_i$ and $c_i=h_i$ $\forall i$ .
Proof: We take the lowest $i$ , $j$ , such that $g_i, h_j >0$ and place a stone at $(i, j)$ , then we subtract $g_i$ and $h_j$ by $1$ each, until $g_i$ and $h_i$ become $0$ $\forall i$ , which will happen when $m$ stones are placed, because $\sum_{i=1}^n g_i$ and $\sum_{i=1}^n h_i$ are both initially $m$ and decrease by $1$ after each stone is placed. Note that in this process $r_i+g_i$ and $c_i+h_i$ remains invariant, thus, the final piling satisfies the conditions above.
By the above lemmas, the number of ways to pile is simply the number of ways to choose the sequences $r_i$ and $c_i$ such that $\sum_{i=1}^n r_i=\sum_{i=1}^n c_i=m$ and $r_i, c_i \geq 0 \forall i$ . By stars and bars, the number of ways is $\binom{n+m-1}{m}^{2}$ .
Solution by Shaddoll