\definestartstop
    [abstract]
    [style=bold,
    after={\blank[big]}]

\starttext
\title{My title: example of a word and character count}

{\strut\tfx Formal guidelines: word count of abstract: $x$, character
count of main text: $x$, character count of figure captions: $x$.}
\blank

\startabstract
This is the abstract. Read this and that.
\stopabstract

% start of the main text
Some random text with formulas

\startformula
y = α + βx + ε, \quad ε \sim N(0,σ^2)
\stopformula

And there are also figures with captions.

\startplacefigure[
    location=bottom,
    title={Residual plot with time $t$ on the x-axis and
    residuals $e_{jt}$ on the y-axis},
    reference={fig:subject-residual}]
\externalfigure[residuals][height=.242\textheight]
\stopplacefigure

And some more text with $x$ and $y$ and $z$, and \placeformula

\startformula \startalign
\NC a =\NC b \NR
\NC c =\NC d \NR
\stopalign \stopformula

And stop.
\stoptext

Try something based on this:

\startluacode
    userdata = userdata or { }

    function userdata.wordcount(listname)
        filename = file.addsuffix(tex.jobname,"words")
        if lfs.isfile(filename) then
            local w = dofile(filename)
            if w then
                if type(w.categories[listname]) == "table" then
                    context(w.categories[listname].total)
                else
                    context(w.total)
                end
                context.par()
            end
        end
    end
\stopluacode
\def\wordcount{%
    \dosingleempty\dowordcount}
\def\dowordcount[#1]{%
    \ctxlua{userdata.wordcount("#1")}}
\setupspellchecking[state=start,method=2]
\ctxlua{languages.words.threshold=1}

\definestartstop
    [abstract]
    [style=bold,
    after={\blank[big]}]

\starttext

    \setupspellchecking[list=abstract]

    \startabstract
        This is the abstract. Read this and that.
    \stopabstract

    \setupspellchecking[list=main]

    Some random text with formulas

    \startformula
        y = α + βx + ε, \quad ε \sim N(0,σ^2)
    \stopformula

    And there are also figures with captions.

    \setupspellchecking[list=figures]

    \startplacefigure[
      location=bottom,
      title={Residual plot with time $t$ on the x-axis and
      residuals $e_{jt}$ on the y-axis},
      reference={fig:subject-residual}]
          \externalfigure[residuals][height=.242\textheight]
    \stopplacefigure

    \setupspellchecking[list=main]

    And some more text with $x$ and $y$ and $z$, and \placeformula

    \startformula \startalign
        \NC a =\NC b \NR
        \NC c =\NC d \NR
    \stopalign \stopformula

    And stop.

    \setupspellchecking[state=stop]

\title{My title: example of a word and character count}

        Abstract: \wordcount[abstract]
        Main: \wordcount[main]
        Figures: \wordcount[figures]
        Wordcount: \wordcount

\stoptext

--
Rik