Proving that if $(X_t)_{t\geq0}$ and $(Y_t)_{t \geq0}$ are continuous and have the same marginal distributions, then $P_\mathbb{X}=P_\mathbb{Y}$.
From the hypothesis and Dynkin's $\pi-\lambda$ theorem [1], you can deduce that the two processes have the same distributions when restricted to rational times. Then continuity gives the desired conclusion.
Without continuity, the following is a well known counterexample: Let $\{X_t\}$ be standard Brownian motion, let $U$ be uniformly distributed in $[0,1]$ and let $\{Y_t\}$ be obtained from $\{X_t\}$ by setting $Y_U=X_U+1$ and $Y_t=X_t$ for all $t \ne U$.
[1] https://en.wikipedia.org/wiki/Dynkin_system