I guess I understand some reasons why we should care for complex structure on manifolds, but what is the reason why product structure is studied? Does it arise naturally somewhere?
The product structure on a smooth manifold $M$ is given by a (1,1)-tensor $E$ such that $E \neq \pm1$ and $E^2 = 1$ and the following integrability condition holds: $$E[X,Y] = [EX,Y] + [X,EY] - E[EX,EY]$$ Compare it to a complex structure $J$ such that $J^2 = -1$ and $$J[X,Y] = [JX,Y] + [X,JY] + J[JX,JY]$$
I'm still studying basic literature, one thing I do know is that on a Lie group it $E$ induces a double algebra structure on a correspoinding Lie algebra, and thus two foliations on the group, if I remember correctly; I guess it is generalized for an arbitrary manifold, so this 'splitting' of the tangent bundle and how it's compatible with other structures (complex, Hermitian, Kähler etc., e.g. a complex product structure is defined by the two respective structures such that $JE = -EJ$) is probably one of the reasons to study it, but I'm not sure this reason is good enough on its own.