How to automatically break long Python module names at dots in \texttt{...} inside tables using \allowbreak or \discretionary?

In your particular case I'd say that only the . is a viable line break for Python modules, so you could also use (in extension to the answers of the suspected duplicate -- this just automatically adds \allowbreak in front of each . in its arguments):

\documentclass[a4paper, 12pt]{report}

\usepackage{tabularx}
\usepackage{array}

\ExplSyntaxOn
\NewDocumentCommand \pymod { m }
  {
    \texttt
      {
        \tl_set:Nn \l_tmpa_tl {#1}
        \tl_replace_all:Nnn \l_tmpa_tl { . } { \allowbreak . }
        \tl_use:N \l_tmpa_tl
      }
  }
\ExplSyntaxOff

\begin{document}
\noindent
\begin{tabularx}{\linewidth}{|X|X|}
  \hline
  \textbf{Modul} & \textbf{Description} \\ \hline
  \pymod{python.package.name.submodul.class.method} & Test\\ \hline
\end{tabularx}
\end{document}

Asides:

• Don't resize tables using \adjustbox or \scalebox or anything like that! It leads to inconsistent font sizes and rule thicknesses and just looks ugly, better use tabularx and X columns than fixed width p columns that you then scale.
• You should really consider whether you want to use H placement. "Floats" placed that way can appear out of order if you use normal floats around them, so as soon as you use a H float you have to manually check every other float.
• Your table would look prettier if you omitted all the vertical rules and used booktabs rules for the horizontal rules (just a \toprule and \midrule around the first two rows, and a \bottomrule at the end, no rules in between).

There is one thing that is exclusive to tables which really makes a difference compared to the suspected duplicate: Inside a table you can use a custom column type. We can use that to automatically insert \pymod around each cell in the table.

In this I also change the used \pymod macro to have an optional argument to specify what to insert in front of the periods in case there is a linebreak. To get that behaviour I changed the \allowbreaks to \discretionarys. We can then use \kern1em for our table columns. This will have the effect that every following element will appear to be indented by 1em. I also use booktabs rules to get a prettier result.

\documentclass[a4paper, 12pt]{report}

\usepackage{booktabs}
\usepackage{tabularx}
\usepackage{array}
\usepackage{collcell}

\ExplSyntaxOn
\NewDocumentCommand \pymod { O{} m }
  {
    \texttt
      {
        \tl_set:Nn \l_tmpa_tl {#2}
        \tl_replace_all:Nnn \l_tmpa_tl { . } { \discretionary{}{#1}{} . }
        \tl_use:N \l_tmpa_tl
      }
  }
\ExplSyntaxOff

\newcolumntype\pymod[1]
  {>{\collectcell{\pymod[\kern1em]}}#1<{\endcollectcell}}

\begin{document}
% showing the behaviour in normal text without trailing indentation.
\begin{minipage}{5cm}
  \pymod{python.package.name.submodul.class.method}
\end{minipage}

\bigskip
\noindent
\begin{tabularx}{\linewidth}{\pymod{X}X}
  \toprule
  % use \multicolumn so that this one cell isn't affected by \pymod
  \multicolumn{1}{X}{\textbf{Modul}} & \textbf{Description} \\
  \midrule
  python.package.name.submodul.class.method & Some really cool Python module
  that is supper handy. In particular this provides a method of a class of a
  submodule. \\
  \addlinespace
  tex.sx.answer.skillmon.smart & Smart answers are just smart, and make tables
  look nicer with less user input \\
  \bottomrule
\end{tabularx}
\end{document}