I assume that the two numbers are on the faces of the dice.
Let $\displaystyle E=\{1,\dots,6\}$ and $\displaystyle \mathcal{E}=2^E$; then $\displaystyle E$ is the domain of $\displaystyle X$. The
definition says that $\displaystyle X:\Omega\to E, X(x,y)=\max(x,y)$ is an $\displaystyle (E,\mathcal{E})$-valued random variable if $\displaystyle X^{-1}(B)\in\mathcal{F}_2$ for all $\displaystyle B\in\mathcal{E}$. That is, for any subset $\displaystyle B$ of values that $\displaystyle X$ may have, the set of outcomes from $\displaystyle \Omega$ that produce an answer in $\displaystyle B$ is a legitimate event, i.e., belongs to the set of events $\displaystyle \mathcal{F}_2$. For this example, this fact is obvious because $\displaystyle \mathcal{F}_2$ includes all possible subsets of $\displaystyle \Omega$.
In contrast, $\displaystyle X$ is not a random variable with respect to $\displaystyle (\Omega,\mathcal{F}_1,\mathbb{P})$ and $\displaystyle (E,\mathcal{E})$. E.g., consider $\displaystyle X^{-1}(\{2\})$.