13. Let the pair of positive integers $(x, y)$ satisfy the condition, then $x^{y}(x y)^{x}=y^{y}$, hence $x^{y} \mid y^{y}$, so $x \mid y$. Let $y=k x, k \in \mathbf{N}^{*}$, then $k^{y}=\left(k x^{2}\right)^{x}$, thus $k^{k}=k x^{2}$, which means $x^{2}=k^{k-1}$. Therefore, all pairs of positive integers $(x, y)$ that satisfy the condition are $\left((2 n+1)^{n},(2 n+1)^{n+1}\right)$ or $\left((2 m)^{4 m^{2}-1},(2 m)^{4 m^{2}+1}\right)$, where $m, n$ are any positive integers.