NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure.
In an answer to the question
I claimed that there exist (in a mathematical sense), Hamiltonian systems on the torus (and in fact on higher genus surfaces as well). However, when pressed to come up with a physical system in the real world (even an idealized one) whose dynamics could be modeled as a Hamiltonian system on the torus, I could not think of one.
Does such a system exist?
I would even be satisfied with a non-classical system which can somehow effectively be described by a Hamiltonian system on the torus, although I'm not sure that the OP of the other question I linked to above would be.