How to create a macro that creates other macros (that takes arguments)?

As @daleif remarked, put\expandafter before \def. And to get \thepointA you have to use \csname ... \endcsname there. I guess this is what you want:

\newcommand*{\newpoint}[2]{%
   \tikzset{#1/.style={#2}}%
   \newcounter{point#1}\setcounter{point#1}{0}%
   \expandafter\def\csname Point#1\endcsname (##1,##2)|##3;{%
        \stepcounter{point#1}\fill[#1] (##1,##2) circle (2pt) node[above](#1-\csname thepoint#1\endcsname){##3};}%
}

And here is a solution without a real counter for each call of \newpoint, based on the solution with a pseudo-counter by @egreg, but with an increment of the pseudo-counter, each time the \point<name> macro is called.

\newcounter{pointnumber}
\newcommand\StepPointNumber[1]{%
    \setcounter{pointnumber}{\csname number@point#1\endcsname}%
    \stepcounter{pointnumber}
    \expandafter\xdef\csname number@point#1\endcsname{\thepointnumber}%
              \expandafter\show\csname number@point#1\endcsname
}
\newcommand*{\newpoint}[2]{%
   \tikzset{#1/.style={#2}}%
   \expandafter\def\csname number@point#1\endcsname{0}%
   \expandafter\def\csname Point#1\endcsname (##1,##2)|##3;{%
        \StepPointNumber{#1}%
        \fill[#1] (##1,##2) circle (2pt) node[above](#1-\csname number@point#1\endcsname){##3-\csname number@point#1\endcsname};}%
}

I don't see the need to allocate a new counter, unless you want to do arithmetic with it.

The main point, though, is that you need to use \csname in the right way:

\documentclass[tikz, border=2mm]{standalone}

\newcommand*{\newpoint}[2]{%
   \tikzset{#1/.style={#2}}%
   \expandafter\def\csname number@point#1\endcsname{0}%
   \expandafter\def\csname Point#1\endcsname (##1,##2)|##3;{%
        \fill[#1] (##1,##2) circle (2pt) node[above](#1-\csname number@point#1\endcsname){##3};}%
}

\newpoint{A}{red}

\begin{document}

\begin{tikzpicture}
  \PointA(1,2)|A;
\end{tikzpicture}

\end{document}