This is a wrong use of closure property.
- DCFL is closed under complement, which means it $\overline{DCFL}$ is always $DCFL$.
- DCFL is not closed under Intersection, which means the resultant language after the intersection of 2 DCFL languages may or may not be DCFL.
Now in this particular example,
$L_1$ is $DCFL$ and $L2 = \overline{L1}$. Hence $L2$ is also $DCFL$
Now, $L= L_1\cap L_2$ can be written as $L= L_1\cap \overline{ L_1} = \phi$
Hence L is regular and also CSL. But more correctly it is Regular.