Align text in minipage at same height

Use [t] as the option -- \begin{minipage}[t]{0.5\textwidth}:

\documentclass{article}
\begin{document}
  \begin{minipage}[t]{0.5\textwidth}
    \begin{flushleft}
        \large \textit{Author}:\vspace*{0.5cm} \\
        Me
    \end{flushleft}
\end{minipage}%
%
\begin{minipage}[t]{0.5\textwidth}
    \begin{flushright}
        \large \textit{Supervisor}:\vspace*{0.5cm} \\
        Him\\
        \vspace*{0.5cm}
        \large \textit{Advisor}:\vspace*{0.5cm} \\
        Other Guy
    \end{flushright}
\end{minipage}%
\end{document}

Just to highlight the jdods user remark in the comments to David Carlisle's solution:

Although the solution given by user David Carlisle is fully functional for various cases, there are some cases because of the extra settings in the document, which do not allow to maintain the alignment, so the use of \ vspace {0pt} is a more complete solution but I believe it does not end for other possible deformations that could arise. The use of \ vspace {0pt} was pointed out by user jdods and therefore does not represent a response of my own. To do so, see https://latex.org/forum/viewtopic.php?t=1059 and consequently https://www.ctan.org/pkg/epslatex.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{fancybox}
\usepackage{geometry}
\thispagestyle{empty}
\begin{document}
\centering    
With \verb|\vspace{0pt}| at the top of each minipage of this example of a pair of minipages see below: %
\\
\fbox{
        \begin{minipage}{0.8\linewidth}\vspace{0pt}%
        \fontsize{10}{6}\selectfont Reference.
    \end{minipage}%
    }%
        \fbox{%
        \begin{minipage}{0.1\linewidth}\vspace{0pt}%
        \fontsize{10}{6}\selectfont Reference.
            \end{minipage}
    }
        \end{document}