Why is string concatenation not distributive over intersection?
When we say that concatenation doesn't distribute over intersection, we don't mean that $L(L_1\cap L_2) \neq LL_1 \cap LL_2$ for all languages $L,L_1,L_2$, but rather that this inequality holds for some $L,L_1,L_2$. Here is one example:
$$
L = \{a,aa\}, L_1 = \{a\}, L_2 = \{aa\}.
$$
In this case $L(L_1 \cap L_2) = \emptyset$ but $LL_1 \cap LL_2 = \{aaa\}$.