We single out a large class of groups
{\rm {\boldsymbol {\mathscr {M}}}} for which the following unique prime factorization result holds: if
\Gamma _1,\ldots,\Gamma _n\in {\rm {\boldsymbol {\mathscr {M}}}} and
\Gamma _1\times \cdots \times \Gamma _n is measure equivalent to a product
\Lambda _1\times \cdots \times \Lambda _m of infinite icc groups, then
n \ge m, and if
n = m, then, after permutation of the indices,
\Gamma _i is measure equivalent to
\Lambda _i, for all
1\leq i\leq n. This provides an analogue of Monod and Shalom's theorem [Orbit equivalence rigidity and bounded cohomology, Ann. of Math. 164 (2006), 825–878] for groups that belong to
{\rm {\boldsymbol {\mathscr {M}}}}. Class
{\rm {\boldsymbol {\mathscr {M}}}} is constructed using groups whose von Neumann algebras admit an s-malleable deformation in the sense of Sorin Popa and it contains all icc non-amenable groups
\Gamma for which either (i)
\Gamma is an arbitrary wreath product group with amenable base or (ii)
\Gamma admits an unbounded 1-cocycle into its left regular representation. Consequently, we derive several orbit equivalence rigidity results for actions of product groups that belong to
{\rm {\boldsymbol {\mathscr {M}}}}. Finally, for groups
\Gamma satisfying condition (ii), we show that all embeddings of group von Neumann algebras of non-amenable inner amenable groups into
L(\Gamma ) are ‘rigid’. In particular, we provide an alternative solution to a question of Popa that was recently answered by Ding, Kunnawalkam Elayavalli, and Peterson [Properly Proximal von Neumann Algebras, Preprint (2022), arXiv:2204.00517].