What is $\mathbb P\{W\leq x,Y\leq y\}$ ?Is it $\mathbb P\{\omega \in \Omega : X(\omega )\leq x,Y(\omega )\leq y\}$?

The correct interpretation is the first one: $\mathbb P\{X\le x,Y\le y\}=\mathbb P\{\omega \in \Omega \mid X(\omega )\leq x,Y(\omega )\leq y\}$. You say

I know that $P(X\in A,Y\in B)$ is a product measure on $\mathbb R^2$,

but this is untrue in general. This is only true when $X$ and $Y$ are independent.

The interpretation $\mathbb P\{(\omega ,\omega ')\in \Omega ^2\mid X(\omega )\leq x,Y(\omega ')\leq y\}$ cannot be correct, since $\mathbb P(A)$ is defined for measurable subsets of $\Omega$, not subsets of $\Omega^2$.

The product measure is in no way involved here. Product measure makes its appearance only when $X$ and $Y$ are independent. The correct expression is $P\{\omega \in \Omega: X(\omega) \leq x,Y(\omega) \leq y\}$.