Is there any combinatorial interpretation or bijective proof for this identity $$2C_n=4{2n \choose n}-{2n+2 \choose n+1}$$ where $C_n$ is the sequence of Catalan numbers?

There is an obvious bijective proof of the identity $$ 2 \binom{2n}{n} + 2 \binom{2n}{n + 1} = \binom{2n+2}{n+1}$$ and also a bijective proof of $$ 2\binom{2n}{n} - 2 \binom{2n}{n+1} = 2C_n,$$ see the paragraph "second proof" in the wiki page. By combining the two, you get a bijective proof of your identity, in the form $$ 4 \binom{2n}{n} =\binom{2n+2}{n+1} + 2C_n .$$

In the wikipedia page on Catalan numbers there is the following:

"There are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers."

I guess it is a good place to look for an answer.