How do I get a cases-like environment with a right brace?

This is just a quick and inelegant attempt. Which engine do you use (I am asking because you load inputenc (usually used for non-unicode, hence pdfLaTeX) and unicode-math (usually for unicode, hence LuaLaTeX or XeLaTeX)? I am not sure whether you want to use the arrows ⇒ or not. They are not seen in the image but in your source code.

\documentclass[a4paper,10pt]{scrartcl}
\usepackage{cancel}%fürs durchstreichen im mathemodus
\usepackage[b]{esvect}
\usepackage{fontspec}%new
%\usepackage[latin1]{inputenc} % Anpassung des Zeichensatzs (Sonderzeichen)
\usepackage[leqno]{mathtools}
\usepackage{unicode-math}%I guess this does not fit to inputenc
\usepackage{polyglossia}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{dsfont}%new for blackboard bold R in Latin Modern.


\newcommand{\bulletspace}{\bullet\hspace{0.5ex}}%new
\DeclarePairedDelimiterX\innerprod[2]{\langle}{\rangle}{#1,#2}%new. This makes stuff easier. You already loaded mathtools. 
\begin{document}


\begin{align*}
    \begin{rcases}
    \parbox[c]{10.7em}{
        %You may work with the itememize as an alternative. 
        %\begin{itemize}
            %\item 
            $\bulletspace g \parallel E \colon\text{Abstand } d \neq 0$\\
            %\item 
            $\bulletspace g \subset E\colon \text{Abstand } d = 0$
        %\end{itemize}
    }
    \end{rcases}
    \Rightarrow%This is not shown in image
    \begin{rcases}
    \parbox[c]{11.5em}{
        $E\colon \innerprod{\vv{n}}{\vv{r}-\vv{r}_0} = 0$\\ 
    $g\colon \vv{r}(t)=\vv{r}_1+t\vv{a}, \; t \in \mathds R  $ % you might want to use a different font for the real numbers like \mathbf R or \BbbR or ℝ
    }
    \end{rcases}
    \Rightarrow%This is not shown in image
    \parbox[c]{7em}{
        {Skalarprodukt:} \\
        $\innerprod{\vv{n}}{\vv{a}}=0 $
    }
\end{align*}
\end{document}

Here's a solution that sets up a dedicated array-like environment called rarray. It uses two instance of rarray and one instance of array. (Aside: The array environment is very basic, but also very flexible and powerful, environment.)

\documentclass{article}
\usepackage{array,amsmath,amssymb}
\usepackage[normalem]{ulem}  % for \uline macro
\newcolumntype{L}{>{\displaystyle}l} % automatic display-style mode
\newenvironment{rarray}{%
   \left.\begin{array}{L @{{}:{}} L @{} }}{%
   \end{array}\right\} }

\usepackage{sectsty}
\subsectionfont{\normalsize\mdseries\noindent\uline} % optional

\begin{document}

\setcounter{section}{3}
\setcounter{subsection}{9}

\subsection{Lagebeziehungen von einer Gerade und \dots}

\[
\renewcommand\arraystretch{1.25}
\begin{rarray}
\bullet\ g\parallel E & \text{Abstand $d\ne0$} \\
\bullet\ g<E          & \text{Abstand $d=0$} % g<E ??
\end{rarray}
\begin{rarray}
 E & \langle \vec{n},\vec{r}-\vec{r}_0 \rangle\\
 g & \vec{r}(t)=\vec{r}_1+t\cdot\vec{a},\ t\in\mathbb{R}
\end{rarray}
\begin{array}{c}
\text{Skalarprodukt:}\\
\langle\vec{n},\vec{a}\rangle=0
\end{array}
\]
\end{document}