Globally defined macro changes numerical value

You can use mathstyle instead of scalerel:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{mathstyle}

\newcommand\foo[1]{%
  \sbox0{$\currentmathstyle o$}%
  \xdef#1{\the\dimexpr\ht0+\dp0\relax}%
  \texttt{\footnotesize\meaning#1}%
}

\begin{document}

$\foo\myheightT$ has value \myheightT

$_{\foo\myheightS}$ has value \myheightS

\end{document}

As long as you recall it within the scope of the current \ThisStyle, it will remember the height (but will lose valid memory of it upon exit from \ThisStyle). Here, I moved the closing brace of the \ThisStyle after the \textrm invocation.

As Gustavo commented, \ThisStyle invokes the standard LaTeX \mathchoice, which builds 4 styles of boxes and figures out which one to set at the last moment. This means that the last box built will be the \scriptscriptstyle box, and residual measurements and global definitions from the \mathchoice will be associated with that one only, once the \ThisStyle scope has been exited.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{scalerel}
\def\foo{%
    \ThisStyle{%
        \setbox0=\hbox{$\SavedStyle o$}%
        \global\edef\myheight{\the\dimexpr\ht0+\dp0\relax}%
        \textrm{Written \string\myheight\space as \meaning\myheight;~}%
    %
    \textrm{reading \string\myheight\space as \meaning\myheight}}%
}
\begin{document}
    $\foo$
\end{document}

However, here is the best that a \mathchoice approach can do...it can save all 4 heights globally in different macros, but it will not remember the particular style that applied at the time, inside the \mathchoice.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{scalerel}
\makeatletter
\def\foo{%
 \ThisStyle{\setbox0=\hbox{$\SavedStyle o$}%
   \expandafter\xdef\csname myheight\m@switch\endcsname{\the\dimexpr\ht0+\dp0\relax}%
   \textrm{Written \string\myheight\space as %
     \expandafter\meaning\csname myheight\m@switch\endcsname;~}%
}}
\makeatother
\begin{document}
$\foo $

$\scriptstyle\foo$

But here are the 4 sizes globally saved:\\
Display style: \myheightD\\
Text style: \myheightT\\
Script style: \myheightS\\
Script-Script style: \myheights
\end{document}

\ThisStyle is implemented through \mathchoice; in practice, this means that its argument is processed four times, one for each of the four possible math styles, and the resulting four math lists are stored away and used later, during the process of converting a math list into its horizontal equivalent, when the actual math style to use becomes known. This entails that each assignment contained in the argument is executed four times too, so that, if it is a global assignment, only the effects of the last one will survive. And since the order of execution is \displaystyle -> \textstyle -> \scriptstyle -> \scriptscriptstyle…