The notation is the following: $G$ is a group acting on a set, $G_x = \{g \in G\mid gx = x \}$. What does $G/G_x$ look like? relevant
In the above equation, is it an abusive of notation to write $G/G_x$ because $G_x$ is not neccesarily normal? If so what does it actually mean?
If $G$ is a group and $H$ its subgroup then
$$G/H:=\{ gH\ |\ g\in G\}$$
is the set of all (left) cosets of $H$ in $G$. This is a well defined set, regardless of whether $H$ is normal or not. Normality only gives us that this set together with $(gH, g'H)\mapsto gg'H$ operation is a well defined group, a.k.a. the quotient group. Nothing else.