That notion of Frobenius morphism between categories is a generalization of Frobenius algebras (those which have a non-degenerate mulplicative bilinear form) to triangulated categories.
This is quite unrelated to the Frobenius morphism on a scheme. There are lot of things named ater Frobenius!
On the other hand, Frobenius categories show up all the time in geometrical contexts. They provide a nice way to construct triangulated caegories (and the triangulated categories so constrcted are particularly nice: they are ´algebraic´) For example, they are used to construct one of the categories equivalent to the derived category of coherent sheaves on projective space in the canonical example of why derived categories are relevant to geometry! This is explained in the book by Gelfan'd and Manin on homological algebra, if I recall correcty.