How much coupling does product measure have? Given two probability measures on the same measurable space $\Omega$, is their product measure the coupling with the biggest probability of $\\{(x, y): \forall x, y \in\Omega, x \neq y\\}$? If not, what is the coupling with the biggest probability of $\\{(x, y): \forall x, y \in\Omega, x \neq y\\}$? How is the amount of coupling quantified for the product measure? Thanks and regards!

No: try both marginals equal to the uniform distribution on a set of size $n$, then the largest probability of the complement of the diagonal is $1$ while it is $1-1/n$ for the product distribution.