Listings \lst@ProcessOther command doesn't work on ")" character when using the breaklines feature. How can I solve this?

Well listings adds special processing instructions for ) to a hook and that breaks special processing of the char:

\lst@AddToHook{SelectCharTable}
    {\lst@ifbreaklines \lst@Def{`)}{\lst@breakProcessOther)}\fi}

You could remove that again and then your code works, but quite probably there is a reason for this and it will change line breaking so you should better report that to the maintainer.

\documentclass{article}
\usepackage{listings}
\usepackage{color}

\makeatletter
\lst@CCPutMacro
    \lst@ProcessOther {"28}{% process "(" character
        \textcolor{red}{(}
    }
    \lst@ProcessOther {"29}{% process ")" character
    \textcolor{red}{)}
    } 
    \lst@ProcessOther {"5B}{% process "[" character
        \textcolor{green}{[}
    }
    \lst@ProcessOther {"5D}{% process "]" character
        \textcolor{green}{]}
    } 
    \lst@ProcessOther {"7B}{% process "{" character
        \textcolor{cyan}{\{}
    }
    \lst@ProcessOther {"7D}{% process "}" character
    \textcolor{cyan}{\}}
    } 
    \@empty\z@\@empty

\usepackage{etoolbox}
\ExpandArgs{c}\patchcmd{\@lst hk@SelectCharTable}
   {\lst@ifbreaklines \lst@Def {`)}{\lst@breakProcessOther )}\fi}
   {}{}{\fail}

\makeatother  

\lstdefinestyle{breakable}{breaklines=true}
\lstdefinestyle{unbreakable}{breaklines=false}
\lstset{frame=single,basicstyle=\ttfamily,}


\begin{document}

\begin{lstlisting}[style=breakable,title={\ttfamily breaklines = true}]
([({([()])})])
\end{lstlisting}

\begin{lstlisting}[style=breakable,title={\ttfamily breaklines = true  without )}]
[{[{[{[]}]}]}]
\end{lstlisting}

\begin{lstlisting}[style=unbreakable,title={\ttfamily breaklines = false}]
([({([()])})])
\end{lstlisting}


\end{document}