Welcome to TeX.SE! The reason why you cannot really fill the regions between the paths in your picture is that you used shorten >=... with some negative dimensions. Hence the paths are shorter than the dashed lines. I fixed that to make it work. And even though you could do that with the pgfplots library fillbetween, this is not necessary here since you only have straight lines. And I used backgrounds and slightly changed the order in which things get drawn in order not to overwrite your planets.
\documentclass[10pt,a4paper,twoside]{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\usetikzlibrary{intersections,backgrounds}
\begin{document}
\begin{tikzpicture}
\fill[color=yellow] (0, 0) circle (2); % Sonne im Ursprung (0, 0)
\shade[right color=lightgray] (7.5, 0) circle (0.5); % Sonne im Ursprung (5, 0)
\draw[dashed, color=red,name path = A] (0, 2) --
node[near start,sloped,above]{\textcolor{red}{Randstrahl}} (10, 0)
coordinate (X1);
\draw[dashed, color=red ,name path = C] (0, 2) --
(10, -1.33)
coordinate (X2);
\draw[dashed, color=red ,name path = B] (0, -2) -- (10, 0)
coordinate (X3);
\draw[dashed, color=red ,name path = D] (0, -2) -- (10, 1.33)
coordinate (X4);
\begin{scope}[on background layer]
\fill[gray,
name intersections={of=A and D,by=X5}] (X3) -- (X5) -- (X4);
\fill[gray,
name intersections={of=C and B,by=X6}] (X1) -- (X6) -- (X2);
\fill[black] (X6) -- (X3) -- (X5);
\end{scope}
\shade[shading=ball, ball color=blue] (10, 0) circle (1); % Erde im Ursprung (10, 0)
\end{tikzpicture}
\end{document}
Here is a version using tangents, as pioneered by esdd in this answer yet with a slightly different implementation/ Specifically, I use this answer to make the dashed lines tangent to both circles. There are other ways to achieve this, most notably those using the tkz-euclide library, see e.g. here. This solution finds the intersections without the intersection library, which works fine for straight lines.
\documentclass[tikz,border=3.14mm]{standalone}
\usetikzlibrary{backgrounds,calc}
\begin{document}
\begin{tikzpicture}
\pgfmathsetmacro{\rsun}{2}
\pgfmathsetmacro{\rmoon}{0.5}
\pgfmathsetmacro{\mid}{\rsun/(\rsun + \rmoon)}
\pgfmathsetmacro{\out}{\rsun/(\rsun - \rmoon)}
\node[circle,draw,inner color=yellow,outer color=orange!50!yellow,minimum
size=2*\rsun*1cm,alias=c1] (Sonne) at (0, 0){}; % Sonne
\node[left color=lightgray,right color=darkgray,circle,draw,minimum
size=2*\rmoon*1cm,alias=c2] (Mond) at (7.5, 0){}; % Mond
% from https://tex.stackexchange.com/a/7209/121799
\path (Sonne.center) -- coordinate[pos=\mid] (mid) (Mond.center);
\path (Sonne.center) -- coordinate[pos=\out] (out) (Mond.center);
\coordinate (Erdmittelpunkt) at (10,0);
\coordinate (Nordpol) at (10,1);
\foreach \i in {Sonne,Mond}
\foreach \j in {1,2}
\foreach \k in {mid,out}
\coordinate (\i-\k-\j) at (tangent cs:node=\i,point={(\k)},solution=\j);
\draw[dashed, color=red] (Sonne-mid-1) --
(intersection cs:first line={(Sonne-mid-1)--(Mond-out-1)},
second line={(Erdmittelpunkt)--(Nordpol)}) coordinate (aux1);
\draw[dashed, color=red] (Sonne-mid-2) --
(intersection cs:first line={(Sonne-mid-2)--(Mond-out-2)},
second line={(Erdmittelpunkt)--(Nordpol)}) coordinate (aux2);
\draw[dashed, color=red] (Sonne-out-1) --
(intersection cs:first line={(Sonne-out-1)--(Mond-mid-1)},
second line={(Erdmittelpunkt)--(Nordpol)}) coordinate (Oberhalb);
\draw[dashed, color=red] (Sonne-out-2) --
(intersection cs:first line={(Sonne-out-2)--(Mond-mid-2)},
second line={(Erdmittelpunkt)--(Nordpol)}) coordinate (Unterhalb);
% auxiliary coordinates
\path(intersection cs:first line={(Sonne-mid-1) --(aux1)},
second line={(Sonne-out-2)--(Unterhalb)}) coordinate (Mond1);
\path(intersection cs:first line={(Sonne-mid-2) --(aux2)},
second line={(Sonne-out-1)--(Oberhalb)}) coordinate (Mond2);
% fills on background in order not to distort Sonne' nor the tangents
\begin{scope}[on background layer]
\fill[gray] (Unterhalb) -- (Mond1) -- (aux1);
\fill[gray] (Oberhalb) -- (Mond2) -- (aux2);
\fill[black] (Mond1) -- (aux1) -- (aux2) -- (Mond2);
\end{scope}
\shade[shading=ball, ball color=blue] (Erdmittelpunkt) circle (1); % Erde im Ursprung (10, 0)
\end{tikzpicture}
\end{document}
