There are many results of this type, and in general they go under the name of minimax theorems. Often some sort of convexity / concavity assumption is made, as in Sion's Minimax Theorem.

You are correct in stating that uniqueness of optimizers gives you equality when $f$ is continuous with domain $[0,1]\times [0,1]$. However, this can fail if the domain of $f$ is different. For example, let $f$ be the arc length metric on the circle. Then $M_x = \pi$ for all $x$, $m_y = 0$ for all $y$. The optima are achieved uniquely at $y = -x$ and $x=y$, respectively, but $\pi = \inf_x M_x \neq \sup_y m_y = 0$.

There are many results of this type, and in general they go under the name of minimax theorems. Often some sort of convexity / concavity assumption is made, as in Sion's Minimax Theorem.

You are correct in stating that uniqueness of optimizers gives you equality when $f$ is continuous with domain $[0,1]\times [0,1]$. However, this can fail if the domain of $f$ is different. For example, let $f$ be the arc length metric on the circle. Then $M_x = \pi$ for all $x$, $m_y = 0$ for all $y$. The optima are achieved uniquely at $y = -x$ and $x=y$, respectively, but $\pi = \inf_x M_x \neq \sup_y m_y = 0$.