It's a timing problem. If I do
\documentclass{article}
\usepackage{microtype}
\usepackage{amsmath}
\ShowCommand\eqref
\begin{document}
\ShowCommand\eqref
\stop
I get
> \eqref=robust macro:
->\protect \eqref .
> \eqref =\long macro:
#1->\textup {\tagform@ {\ref {#1}}}.
<argument> \eqref
l.5 \ShowCommand\eqref
?
(/usr/local/texlive/2025/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def)
(./pardon.aux)
(/usr/local/texlive/2025/texmf-dist/tex/latex/microtype/mt-cmr.cfg)
> \eqref=robust macro:
->\protect \eqref .
> \eqref =\long macro:
#1->\textup {\@nameuse {MT@patch@saved@\string \tagform@ }{\ref {#1}}}.
<argument> \eqref
l.9 \ShowCommand\eqref
?
So you see that microtype changes the definition of \eqref at begin document and you should do the redefinition (I guess you don't just want to get an alias to \eqref) at begin document.
Ensure to do
\AtBeginDocument{\NewCommandCopy{\stdeqref}{\eqref}}
in order to have an alias so you can redefine \eqref to your liking in terms of \stdeqref. Using \let on a robusted command is, to say the least, risky.
Some more details
If you do \let\stdeqref\eqref, the meaning of \stdeqref becomes \protect\eqref• (where the bullet means a space in the control sequence name). As we've seen, microtype redefines \eqref• and so, if you call
\stdeqref{a}\allowbreak\stdeqref
you get a result that's exactly the same as \eqref{a}\allowbreak\eqref{a}. Indeed, this is what happens:
\documentclass{article}
\usepackage{microtype}
\usepackage{amsmath}
\let\stdeqref\eqref
\begin{document}
\begin{equation}\label{a}
A
\end{equation}
\eqref{a}\allowbreak\eqref{a}
\stdeqref{a}\allowbreak\stdeqref{a}
\end{document}
To the contrary, if you do \NewCommandCopy{\stdeqref}{\eqref}, you get
Why? Because \NewCommandCopy defines \stdeqref to be \protect\stdeqref• and copies \eqref• into \stdeqref•, so the microtype patch isn't applied when you use \stdeqref{a}.
Of course this is wrong and, as said, you should do
\AtBeginDocument{%
\NewCommandCopy{\stdeqref}{\eqref}%
% the redefinition of \eqref using \stdeqref goes here
}