$\Box$ and $\Diamond$ are dual in the natural extension to modal logic of the De Morgan duality from classical propositional logic, which is itself essentially a logical manifestation of the fact that in the two-element Boolean algebra (or any Boolean algebra for that matter) $\langle B,\le,\land,\lor,\neg,0,1\rangle$, $\neg$ is an isomorphism of $B$ to its dual algebra $\langle B,\ge,\lor,\land,\neg,1,0\rangle$. (That is, in the modal case, a similar dual isomorphism applies to modal algebras, i.e., Boolean algebras with operators.)

This meaning of “dual” as order-reversing isomorphism (or the duality in elementary geometry, for another example) has been here long before anyone invented category theory and dual categories, although I do not doubt that one can make the two-element Boolean algebra into a category to explain in that way.