If $L$ is regular then $L^{|2|}=\{w_1w_2 \mid w_1,w_2\in L, |w_1|=|w_2|\}$ is context-free
Let $\Sigma' = \{\sigma' : \sigma \in \Sigma\}$ be a tagged version of the alphabet of $\Sigma$. Let $L'$ be a version of $L$ obtained by tagging all letters. Let $h\colon \Sigma \cup \Sigma' \to \Sigma$ be a homomorphism which removes tags. Then
$$
L^{|2|} = h(LL' \cap \{xy : x \in \Sigma^n, y \in \Sigma^{\prime n} \text{ for some } n\}).
$$
Standard closure properties of context-free languages imply that $L^{|2|}$ is context-free.