How to implement \ifxcase: case equivalent of \ifx

I'd use a different syntax (a hash table lookup rather than a nested switch) that allows the coloured symbols to be declared separately, potentially at different points in the code.

\documentclass{article}
\usepackage{mathtools}

\usepackage{xcolor}

\newcommand*{\ColorSymbol}[1]{%
  \expandafter\ifx\csname CS-\string#1\endcsname\relax
  #1\footnote{Unknown symbol: \string#1}%
  \else
  \csname CS-\string#1\expandafter\endcsname
  \fi}

\newcommand\defColorSymbol[3][]{%
    \expandafter\def\csname CS-\string#2\endcsname{%
    #1{\color{#3}#2}}}

\defColorSymbol\exists{red}
\defColorSymbol[\mathrel]\ge{orange}
\defColorSymbol[\mathrel]\le{blue}



\begin{document}

$\ColorSymbol\exists x$

$A \ColorSymbol= B$

$a \ColorSymbol\ge n$,
$x \ColorSymbol\le z$

\end{document}

You're loading xstring that already provides the feature:

\documentclass{article}
\usepackage{xcolor}
\usepackage{xstring}

\newcommand{\ColorSymbol}[1]{%
  \IfStrEqCase*{#1}{%
    {\exists}{\textcolor{red}{#1}}%
    {\ge}{\mathrel{\textcolor{orange}{#1}}}%
    {\le}{\mathrel{\textcolor{blue}{#1}}}%
  }[#1\footnote{Unknown symbol: \string#1}]%
}

\begin{document}

$\ColorSymbol\exists x$

$A \ColorSymbol= B$

$a \ColorSymbol\ge n$,

$x \ColorSymbol\le z$

\end{document}

With xparse it's similar:

\documentclass{article}
\usepackage{xcolor}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\ColorSymbol}{m}
 {
  \str_case:nnF { #1 }
   {
    {\exists}{\textcolor{red}{#1}}
    {\ge}{\mathrel{\textcolor{orange}{#1}}}
    {\le}{\mathrel{\textcolor{blue}{#1}}}
   }
   {#1\footnote{Unknown symbol: \string#1}}
 }
\ExplSyntaxOff

\begin{document}

$\ColorSymbol\exists x$

$A \ColorSymbol= B$

$a \ColorSymbol\ge n$,

$x \ColorSymbol\le z$

\end{document}

Yet another option, here's an approach loosely based on C syntax.

\documentclass{article}
\usepackage{mathtools}
\usepackage{xstring}
\usepackage{xcolor}

\makeatletter
    \newcommand\switch[3][\ifx]{%provide switch-case structure with an if
        \def\@parsed@ifs{}%contains the arguments recast to standard \if\else\fi
        \def\@allfis{}%contains all of the \fis
        \def\case@with@option[##1]##2##3{%process a case with optional relation ##1
            \g@addto@macro\@allfis{\fi}%
            \g@addto@macro\@parsed@ifs{#1#2##1##2##3\else}}%            
        \def\case{\@ifnextchar[{\case@with@option}{\case@with@option[]}}%
        \def\default##1{\g@addto@macro\@parsed@ifs{##1}}%
        #3\expandafter\@parsed@ifs\@allfis%
    }
\makeatother

%%% Would prefer some case like syntax:
\newcommand*{\ColorSymbol}[1]{%
    \switch{#1}{%
        \case{\exists}{{\color{red}#1}}%
        \case{\ge}{\mathrel{\color{orange}#1}}%
        \case{\le}{\mathrel{\color{blue}#1}}%
        \default{#1\footnote{Unknown symbol: \string#1}}}
}%

\begin{document}

$\ColorSymbol\exists x$

$A \ColorSymbol= B$

$a \ColorSymbol\ge n$,
$x \ColorSymbol\le z$

\end{document}

The conditional is left as an option (\if, \ifx, \ifnum, and \ifdim all work). To support those using a relation, \case is defined with an optional parameter. For example, two is the result from:

\def\testnumber{2}%
\def\anothernumber{5}%
\switch[\ifnum]{\testnumber}{%
    \case[=]{0}{zero}%
    \case[=]{1}{one}%
    \case[=]{2}{two}%
    \case[=]{\anothernumber}{another number}%
    \default{not 0, 1, or 2}}

Obviously this will fail spectacularly if \case or \default are defined elsewhere and used as arguments in the \case and \default statements (e.g., \case{\exists}{\default\case}). For the same reason, nesting \switch calls would be problematic as well. As such, this approach is limited.