It's not context free for the same reason that $L' = \{w w \mid w \in \Sigma^*\}$ is not context free, and the proof below is a simple adaption of the standard proof for the language $L'$, using the pumping lemma:
Choose the string $0^p1^p1^p0^p0^p1^p1^p0^p$, (which is $\omega \omega^r \omega \omega^r$ where $\omega = 0^p 1^p$). [Note 1] Now any $vwx$ whose length is no greater than $p$ is either entirely within the first half of the string, entirely within the second half of the string, or entirely within the central span of $0^{2p}$. Repeating $v$ and $x$ 0 times -- that is, changing $vwx$ to $w$ -- must make the first half of the string different from the second half, either because only one of the halves is altered, or because changing the suffix of one half and the prefix of the other half renders the two halves distinct. So the result is not in $L$.
Since for any $p$, we can find a string which cannot be pumped, we can conclude that there is no pumping length $p$ for the language. That contradicts the pumping lemma, which states that every context-free language has a pumping length. So the language is not context-free.
You can't “take the first $ww^r$ as $v$” because of the restriction that the length of $vwx$ is no greater than the pumping length $p$. So, whatever $p$ is (if it exists) there is some $ww^r$ which is longer, and for the corresponding string $ww^rww^r$ in $L$, $v$ cannot be $ww^r$.
Notes:
- I could have chosen a shorter string, such as $\omega = 0^m1^n$ where $m>0, n>0, 4mn>=p$. But there are no prizes for parsimony here; the string I chose makes the proof less verbose.
