documentclass{standalone}
usepackage{tikz}
usetikzlibrary{angles}

begin{document}
begin{tikzpicture}
coordinate (P) at (0,0);
coordinate (A) at (90+45:1);
coordinate (P') at ([yshift=1cm] P);

fill[blue] (-1,-1) rectangle (1,2);

draw (P) -- (P') (P) -- (A);

pic[draw, pic text=$beta$, angle eccentricity = 1.4] {angle = P'--P--A};
end{tikzpicture}
end{document}

label={[fill=white, fill opacity=.5, text opacity=1] $beta$}

documentclass{standalone}
usepackage{tikz}
usetikzlibrary{angles}

begin{document}
begin{tikzpicture}
coordinate (P) at (0,0);
coordinate (A) at (90+45:1);
coordinate (P') at ([yshift=1cm] P);

fill[blue] (-1,-1) rectangle (1,2);

draw (P) -- (P') (P) -- (A);

pic[draw,
  pic text=$beta$,
  pic text options={fill=white,fill opacity=.5,inner sep=1pt},% <- added
  angle eccentricity = 1.4] {angle = P'--P--A};
end{tikzpicture}
end{document}