unicode-math/mathspec interferes with siunitx

As explained in an answer by nlogax to microtype + siunitx and \micro - mysterious warnings Unicode has—besides the Greek Small Letter Mu character µ—a special Micro Sign µ.

That is to be used here. For some reason this is not present in unicode-math.
I assume that you also use TeX Gyre Pagella as your text font so we can use it from there with \text.

Code

\documentclass{standalone}
\usepackage{unicode-math}
\setmainfont{TeX Gyre Pagella}
\setmathfont{TeX Gyre Pagella Math}
\usepackage[mode = math]{siunitx}
\sisetup{math-micro=\text{µ},text-micro=µ}
\begin{document}
\SI{10}{\micro\m}
\SI[mode=text]{10}{\micro\m} % just for reference
\end{document}

Output

This seems to work

\documentclass{standalone}
\usepackage{unicode-math}
\setmainfont{TeX Gyre Pagella}
\setmathfont{TeX Gyre Pagella Math}
\usepackage[mode = math]{siunitx}
\def\SIUnitSymbolMicro{\textup{μ}}
\begin{document}


\SI{10}{\micro\m}
\end{document}

Note I needed to specify a text font as well.