This compactness property is false in general never true, even for collections of $F_\sigma$ subsets of an uncountable Polish spacesspace. One way to see this is to fix your favorite example of an $F_\sigma$ graph $G$ with clique number $\aleph_1$ and a maximal $G$-clique $K$. Then let your family be $\{G_x : x \in K\}$, where $G_x$ is the set of neighbors of $x$ in $G$.
This compactness property is false in general for collections of $F_\sigma$ subsets of uncountable Polish spaces. One way to see this is to fix your favorite example of an $F_\sigma$ graph $G$ with clique number $\aleph_1$ and a maximal $G$-clique $K$. Then let your family be $\{G_x : x \in K\}$, where $G_x$ is the set of neighbors of $x$ in $G$.