Matrix equality up to row/column permutations problem name
After a few observation in the comments (thanks to Hendrik Jan, Yuval Filmus, and D.W), I post an extended comment as an answer.

It seems (unless I'm missing something) that the problem is equivalent to Hypergraph isomorphism (HI):


to prove that it is equivalent to HI, it is enough to use incidence matrices: every column represents a node of the hypergraph, every row represents an hyperedge (the $1$s are placed on the columns corresponding to the nodes of the hyperedge).



  The question remains valid ... is there a short name for the problem other than Hypergraph Isomorphism (if it appears somewhere not related to graph theory) ?


(if nobody post another answer in a few days I'll temporarily accept mine).

Additional stuff:


for proving that it is GI-complete it is enough to consider the incidence matrix of the source graphs $G_1, G_2$;
the reverse reduction is similar to the well known reduction from HI to GI: it is enough to consider the incidence matrix of the source hypergraphs $H_1, H_2$, transform them to graphs: every column becomes a node $c_i$, every row becomes a node $r_i$; an edge $(c_i,r_i)$ is added if and only if the corresponding cell contains a $1$; and finally a new triangle is linked to every set of nodes $\{c_{i_1},...,c_{i_m}\}$ that represents an hyperedge to preserve hyperedge mapping.