\text{ } subscript size in tikzmath macro is not correct

Add a \noexpand can fix it directly. But I would think the robustify approach makes more sense if this is an ongoing issue.

\documentclass[]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{math}

%%%To fix the text subscript
%\usepackage{etoolbox}
%\makeatletter
%\robustify{\text}
%\makeatother 

\begin{document}
    \begin{tikzpicture}

    \node at (0,0) {$\text{b}_\text{b}$: Correct};

    \tikzmath{let \test=\text{b}_{\text{b}};}
    \node at (0,-1cm) {$\test$: Not Correct};

    \tikzmath{let \test=\text{b}_{\noexpand\text{b}};}
    \node at (0,-2cm) {$\test$: Corrected with noexpand};

    \end{tikzpicture}
\end{document}

You can see what's going on if you add \show\text after the \tikzmath declaration:

> \test=macro:
->\protect \unhbox \voidb@x \hbox {b}_{\protect \unhbox \voidb@x \hbox {b}}.

Indeed, the definition of \text is found in amstext.sty:

% amstext.sty, line 28:
\DeclareRobustCommand{\text}{%
  \ifmmode\expandafter\text@\else\expandafter\mbox\fi}

You also have to know that \tikzmath does full expansion; since \text is not found in math mode, it just does \mbox. Since at the time of let the meaning of \protect is \relax, it goes on untouched; then the conditional is expanded and since TeX is not in math mode, you get \mbox{b}, which becomes

\leavemode\hbox{b}

and finally \unhbox\voidb@x\hbox{b}. This should explain the output of \show above.

Possibly you want to evaluate something instead of having b in the argument to \text; in this case you need to be careful about what to fully expand and what not.

\tikzmath{let \test=\noexpand\text{b}_{\noexpand\text{b}};}

will do, but if instead of b you have something like \textbf{abc}, this will die horribly anyhow.

You need \noexpand in front of both occurrences of \text, in order to suppress its expansion and get the right version when \test is indeed used in math mode.