Choose columnsep depending on page

You tell TeX in the beginning that if this is not a chapter page, it should patch the command @outputdblcol in one way, and otherwise in another. However, in the preamble this is not a chapter page. You need to make TeX check for every single page. To this end I moved \ifchapterpage into the TikZ picture. Also you never set the conditional to false. I added this reset, too. Finally, the arguably more appropriate way to make a tikzpicture take no space is to add overlay.

\documentclass[11pt,a5paper,twocolumn,twoside]{book}

\usepackage{geometry}
    \geometry{
        paper = a5paper,
        layout = a5paper,
        layoutsize = {148 mm, 210 mm},
        includehead = true,
        includefoot = true,
        inner = 1.5 cm,
        outer = 1 cm,
        top = 1.5 cm,
        bottom = 1.5 cm,
        columnsep = .8cm,
    }


\usepackage{tikz}

\newif\ifchapterpage

\usepackage{etoolbox}
    \makeatletter
        %
        \patchcmd\@outputdblcol
        {\normalcolor\vrule\@width\columnseprule}
        {\raisebox{.5\textheight}{%
            \begin{tikzpicture}[overlay]
                %\useasboundingbox;
                \ifchapterpage
                  %\message{page \thepage\space is a chapter page}
                  \fill[blue] (0,0) circle (.1cm);
                  \draw[blue] ++(90:.75) to ++(90:2);
                  \draw[blue] ++(-90:.75) to ++(-90:6);
                  \global\chapterpagefalse
                \else
                  %\message{page \thepage\space is not a chapter page}
                  \fill[blue] (0,0) circle (.1cm);
                  \draw[blue] ++(90:.75) to ++(90:6);
                  \draw[blue] ++(-90:.75) to ++(-90:6);             
                  \fi
            \end{tikzpicture}}}%\mydrawcolumnseprule}
        {}{}
    \makeatother

\usepackage{fancyhdr}
    \pagestyle{fancy}
    \fancyhead{}
    \fancyhead[L]{\rightmark}
    \fancyhead[C]{\thepage}
    \fancyhead[R]{\chaptername}
    \fancyfoot{}

\usepackage{titlesec}
    \titleformat{\chapter}% command
        [block]% shape
        {\centering\itshape\huge}% format
        {}% label
        {0cm}% sep
        {{\normalsize The tale of\\}}% before-code
        [\global\chapterpagetrue\vspace{1cm}]% after-code
    \titlespacing{\chapter}% command
        {0pt}% left margin
        {0pt}% Vertical space before title
        {1cm}% Vertical space between title and text

\usepackage{lipsum}
\chapterpagetrue

\begin{document}

    \chapter*{Rhotingger}

    \lipsum[1-10]

    \chapter*{Blauinger}

    \lipsum[11-20]

\end{document}

Here is a version which uses tikzpagenodes and everypage. Note that \@outputdblcol is not used by paracol or multicol. I created \mycolumnsep in order to test it before adding it to every page.

Don't forget to run it twice, due to [remember picture].

BTW, you can also use this approach to replace fancyhdr.

\documentclass[11pt,a5paper,twocolumn,twoside]{book}

\usepackage{geometry}
    \geometry{
        paper = a5paper,
        layout = a5paper,
        layoutsize = {148 mm, 210 mm},
        includehead = true,
        includefoot = true,
        inner = 1.5 cm,
        outer = 1 cm,
        top = 1.5 cm,
        bottom = 1.5 cm,
        columnsep = .8cm,
    }


\usepackage{tikzpagenodes}
\usepackage{everypage}

\newif\ifchapterpage

\newcommand{\mycolumnsep}{\begin{tikzpicture}[overlay, remember picture]
                %\useasboundingbox;
                \ifchapterpage
                  %\message{page \thepage\space is a chapter page}
                  \fill[blue] (current page text area.center) circle (.1cm);
                  \draw[blue] (current page text area.center) ++(90:.75) to ++(90:2);
                  \draw[blue] (current page text area.center) ++(-90:.75) to ++(-90:6);
                  \global\chapterpagefalse
                \else
                  %\message{page \thepage\space is not a chapter page}
                  \fill[blue] (current page text area.center) circle (.1cm);
                  \draw[blue] (current page text area.center) ++(90:.75) to ++(90:6);
                  \draw[blue] (current page text area.center) ++(-90:.75) to ++(-90:6);             
                  \fi
\end{tikzpicture}}%\mydrawcolumnseprule}
\AddEverypageHook{\mycolumnsep}

\usepackage{fancyhdr}
    \pagestyle{fancy}
    \fancyhead{}
    \fancyhead[L]{\rightmark}
    \fancyhead[C]{\thepage}
    \fancyhead[R]{\chaptername}
    \fancyfoot{}

\usepackage{titlesec}
    \titleformat{\chapter}% command
        [block]% shape
        {\centering\itshape\huge}% format
        {}% label
        {0cm}% sep
        {{\normalsize The tale of\\}}% before-code
        [\global\chapterpagetrue\vspace{1cm}]% after-code
    \titlespacing{\chapter}% command
        {0pt}% left margin
        {0pt}% Vertical space before title
        {1cm}% Vertical space between title and text

\usepackage{lipsum}
\chapterpagetrue

\begin{document}

    \chapter*{Rhotingger}

    \lipsum[1-10]

    \chapter*{Blauinger}

    \lipsum[11-20]

\end{document}