How to change fill opacity without affecting the angle label?

Tikz distinguishes between fill opacity and text opacity. If not both are set, defining fill opacity overrides the setting for text opacity. If you set text opacity=1 you'll get:

Code:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{quotes,angles}

\begin{document}

\begin{tikzpicture}
    \draw (0.4,-0.5) -- (0.4,0.8); % a line to show if opacity works
    \draw coordinate (a) at (3,-1);
    \draw coordinate (b) at (0,0);
    \draw coordinate (c) at (1,1);
    \draw (a) -- (b) -- (c) pic ["$\alpha$",draw=orange,fill=orange!50,fill opacity=0.5, text opacity=1,angle radius=1cm] {angle=a--b--c};
\end{tikzpicture}

\end{document}

You can add options for the label itself using the syntax

"some text" {<options for the label>}, <other options>

hence pic ["$\alpha$" opacity=1,draw=orange,fill=orange!50,fill opacity=0.5,angle radius=1cm] {angle=a--b--c} gives you the desired effect. Note no comma after " ... ". The braces aren't needed in this specific case because there is only one option, opacity.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{quotes,angles}

\begin{document}

\begin{tikzpicture}
    \draw (0.4,-0.5) -- (0.4,0.8); % a line to show if opacity works
    \draw coordinate (a) at (3,-1);
    \draw coordinate (b) at (0,0);
    \draw coordinate (c) at (1,1);
    \draw (a) -- (b) -- (c) pic ["$\alpha$" opacity=1,draw=orange,fill=orange!50,fill opacity=0.5,angle radius=1cm] {angle=a--b--c};
\end{tikzpicture}

\end{document}