i would do the following:
- use
tabularx for table environment
- for the second column use
\renewcommand\tabularxcolumn[1]{m{#1}}, which vertically center cells contents
- include images in table with macro
\adjustimage from the package {adjustbox} move image baseline to its vertical center
- with
\adjustboxset define image size, vertical position and its margins:
with the package cellspace add some vertical space above and below cells with texts
\documentclass{article}
\usepackage{adjustbox}% [demo]
\usepackage{cellspace, tabularx}
\setlength\cellspacetoplimit{3pt}
\setlength\cellspacebottomlimit{3pt}
\addparagraphcolumntypes{X}
%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%
\begin{document}
\begin{table}
\renewcommand\tabularxcolumn[1]{m{#1}}
\adjustboxset{width=.1\textwidth,valign=c, margin=0pt 3pt 0pt 3pt}
\begin{tabularx}{\linewidth}{|c|X|}
\hline
Symbol & Description \\
\hline
\adjustimage{}{example-image-duck} & Text in only one line. \\
\hline
\adjustimage{}{example-image-duck} & Here is a longer text that is displayed over more than one line. The symbol is not vertically centered, but has the same distance to the top as the icon in the first line. Here is a longer text that is displayed over more than one line. The symbol is not vertically centered, but has the same distance to the top as the icon in the first line. \\
\hline
\end{tabularx}
\end{table}
\end{document}

(red lines indicate text borders=
if you like to change size of the particular image, than include width of this image for example as follows:
\adjustimage{width=2cm}{example-image-duck}

edit:
you also can stick with your solution. in preamble just add array package and instead of p{...} column type use m{...}:
\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{array}
\newcommand{\icon}[2][1cm]{\begin{minipage}{#1}\includegraphics[width=#1]{#2}\end{minipage}}
\begin{document}
\begin{tabular}{|c|m{13cm}|}\hline
Symbol & Description \\\hline
& \\ [-0.5em] \icon{icon_settings.pdf} & Text in only one line. \\ [-0.5em] & \\\hline
& \\ [-0.5em] \icon{icon_administrator.pdf} & Here is a longer text that is displayed over more than one line. The symbol is not vertically centered, but has the same distance to the top as the icon in the first line. Here is a longer text that is displayed over more than one line. The symbol is not vertically centered, but has the same distance to the top as the icon in the first line. \\ [-0.5em] & \\\hline
\end{tabular}
\end{document}
