Short horizontal and vertical bar of same length in math

You need to take care of the current math style:

\documentclass{article}
\usepackage{amsmath,amssymb,graphicx}

\newcommand{\vrt}{\shortmid}
\makeatletter
\newcommand{\hrz}{{\mathpalette\hrz@\relax}}
\newcommand{\hrz@}[2]{%
  \mspace{-1mu}%
  \rotatebox[origin=c]{90}{$\m@th#1\,\vrt$}%
  \mspace{-1mu}%
}
\makeatother

\begin{document}

$I^\hrz I^\vrt$

\end{document}

The reason why the rotated \shortmid seems larger is because the default math size setting within \rotatebox would be \textstyle, while it should be \scripstyle if you're using it in a superscript.

You could try with the following definitions of \vertical and \horizontal:

\documentclass{article}

\usepackage{amssymb,graphicx}

\newcommand{\horizontal}[1]{%
  #1^{\rotatebox[origin=c]{90}{$\scriptstyle\shortmid$}}
}
\newcommand{\vertical}[1]{%
  #1^{\shortmid}
}

\begin{document}

$\vertical{I}\ \horizontal{I}$

\end{document}

Preserves the proper math style.

\documentclass{article}
\usepackage{amssymb,graphicx,scalerel}
\newcommand{\vrt}{\shortmid}
\newcommand{\hrz}{{\ThisStyle{\rotatebox[origin=c]{90}{$\SavedStyle\vrt$}}}}
\begin{document}
$I^\hrz I^\vrt$

$\scriptstyle I^\hrz I^\vrt$
\end{document}