Why the euclid distance between the plates of the capacitor has been doubled to find out the capacitance?
I'll just give a hint for the moment.
If we wrote it as
$$ C_0 := \frac 1 2 \frac {\epsilon _0 S} {d} $$
$$ C_1 := \frac 1 2 \frac {\epsilon  S} {d} $$
would it make more sense? Can you see where \$ \frac 1 2 \$ would come in?