How to enforce minimal height in pdfmarkupcomment?

You can patch \pdfmarkupcomment so that every appearance of #2 becomes \strut#2

\documentclass{article}
\usepackage{pdfcomment}
\usepackage{regexpatch}

\makeatletter
\xpatchcmd*{\pdfmarkupcomment}
 {#2}
 {\strut#2}
 {}{}
\makeatother

\begin{document}
  \pdfmarkupcomment[markup=Highlight]{aa}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{AA}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{gg}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{Ag}{Highlight}
\end{document}

but beware that this will apply \strut everywhere, not only when some markup comments are near to each other.

Alternative patching:

\documentclass{article}
\usepackage{pdfcomment}
\usepackage{letltxmacro}

\LetLtxMacro\originalpdfmarkupcomment\pdfmarkupcomment
\renewcommand{\pdfmarkupcomment}[3][]{%
  \originalpdfmarkupcomment[#1]{\strut#2}{#3}%
}

\begin{document}
  \pdfmarkupcomment[markup=Highlight]{aa}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{AA}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{gg}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{Ag}{Highlight}
\end{document}

You might use the height of capital letters and the depth of y, but beware that capital letters usually overshoot:

\documentclass{article}
\usepackage{pdfcomment}
\usepackage{letltxmacro}

\LetLtxMacro\originalpdfmarkupcomment\pdfmarkupcomment
\renewcommand{\pdfmarkupcomment}[3][]{%
  \originalpdfmarkupcomment[#1]{%
    \hbox{\vrule height \fontcharht\font`A depth \fontchardp\font`y width 0pt}%
    #2%
  }{#3}%
}

\begin{document}
  \pdfmarkupcomment[markup=Highlight]{aa}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{AA}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{gg}{Highlight}
  \pdfmarkupcomment[markup=Highlight]{Ag}{Highlight}
\end{document}

You can fix the overshoot by using 1.1\fontcharht\font`A and the result would be

and you may want to do similarly for the depth.

You can simply use \strut to get the height of a base line. Or define your own \boxheight as in the example. You must define it in a macro without argument, otherwise it will not survive the SOUL parser used by pdfcomment.

\documentclass{article}
\usepackage{pdfcomment}
\newcommand\pdfstrutmc[3][\strut ]%
{%
  \pdfmarkupcomment{#1#2}{#3}%
}%
\newcommand\boxheight{\raisebox{-3pt}{\rule{0pt}{15pt}}}
%
\begin{document}
  \pdfstrutmc{aa}{Highlight}
  \pdfstrutmc{AA}{Highlight}
  \pdfstrutmc{gg}{Highlight}
  \pdfstrutmc{Ag}{Highlight}

  \bigskip
  \pdfstrutmc[\boxheight]{Aa}{Highlight}
  \pdfstrutmc[\boxheight]{AA}{Highlight}
  \pdfstrutmc[\boxheight]{gg}{Highlight}
  \pdfstrutmc[\boxheight]{Ag}{Highlight}
\end{document}