Contradiction in regularity of a language
There is a difference between $L_2 = \{ww\mid w \in L_1\}$ and $L_3 = L_1 L_1$. $L_3$ is $\{wx\mid w, x \in L_1\}$; in, other words all strings consisting of two sentences from $L_1$, which is regular / context-free if $L_1$ is regular / context-free. 

$L_2$ is the set of all strings consisting of one repeated string from $L_1$, which is not in general context-free. (It is regular if $L_1$ is finite or is a regular language over a singleton alphabet. But your example does not fall into either of these categories.)