I hope that this does what you want.
% appnameprob.tex SE 553677
\documentclass{article}
\usepackage{blindtext}
\usepackage{tocloft} % useful for controlling the ToC etc
\begin{document}
\tableofcontents
\section{First section}
\blindtext
\section{Second section}
\blindtext
\appendix
\renewcommand{\thesection}{Anhang \Roman{section}} % change the numbering scheme
\addtocontents{toc}{\setlength{\cftsecnumwidth}{6em}} % increase the space in the TOC for the number
\section{The first appendix} \label{appone}
\blindtext
The next appendix is \ref{apptwo}.
\section{The second appendix} \label{apptwo}
\blindtext
The first appendix was \ref{appone}.
\end{document}
It is a modified version of your MWE, for which thank you.
Following your and Donald Arseneau's comments I have added a second answer using the appendix package.
% appnameprob2.tex SE 553677
\documentclass{article}
\usepackage{blindtext}
\usepackage{appendix}
\begin{document}
\tableofcontents
\section{First section}
\blindtext
\section{Second section}
\blindtext
% the next commands are from the appendix package
\renewcommand{\appendixname}{Anhang}
\appendixtitleon % puts \appendixname before each appendix title
\appendixtitletocon % puts \appendixname before each TOC entry
\begin{appendices}
\renewcommand{\thesection}{\Roman{section}} % change the numbering scheme
\section[]{The first appendix} \label{appone} % empty title in the TOC
\blindtext
The next appendix is \ref{apptwo}.
\section[]{The second appendix} \label{apptwo}
\blindtext
The first appendix was \ref{appone}.
\end{appendices}
\end{document}
Depending on your end goal you might want to use bits from both answers (the \ref results differ between the two answers).