How can the stabilizer of a group element have more than one element itself?

The problem is that $g^{-1}g$ is not necessarily $e$! The action of $G$ on itself may be very different from the group law (think about conjugation, for example : in this case, $g^{-1}.g=g^{-1}gg=g\neq e$ in general).

The stabilizer is associated with a group action. Actions aren't guaranteed to have inverses, so acting on the right by $g^{-1}$ is nonsensical.

In general, given some group $G$ acting on a set $X$, the stabilizer is the elements of $G$ that act as the identity on $X$. Consider the action given by $h\mapsto ghg^{-1}$ (conjugation by $g$). This will be invariant under the action if $ghg^{-1}=h$, or if $gh=hg$, so if $g$ commutes with $h$. It follows that the center of $G$ stabilizes the above action.