looking for a combinatorial interpretation given positive integers $n,m$ does the fraction
$$
\frac{(nm)!}{n!^mm!}
$$
count something? Namely does it correspond to the number of possibilities to do something?

Yes, there is a simple possibility to show this by counting the cosets of certain symmetric groups.
Consider the symmetric group of $n$ elements $S_n$. Since $S_n ≀ S_m$ can obviously interpreted as a subgroup of $S_{nm}$, you see that $\frac{(nm)!}{n!^m m!}$ is the number of cosets of $S_n ≀ S_m$ in $S_{nm}$.
