Does energy of a capacitor means energy stored in both plates?

It is the total energy stored in the entire system.

Consider that you have an uncharged parallel plate capacitor and you decide to charge it up such that the charge on one plate is $+Q$ and the charge on the other plate is $-Q$ where $Q>0$. At any point in time, the charge on the positive plate is $q$, where $0leq qleq Q$, which gives a potential difference between the plates of $frac{q}{C}$.

We know that $dW=Vdq$ and so $$W=frac{1}{C}int_0^Qqdq=frac{Q^2}{2C}$$ As you can see, the derivation of the formula above utilized the potential difference between both plates and so the state of the second plate was also taken into account. Therefore, $frac{Q^2}{2C}$ is the total energy of the entire system.