Well, yes, the amount of energy stored goes up as the square of voltage for a given capacitance. However, for a given dielectric getting twice the voltage requires twice the thickness and cuts the charge in half -- so the energy per unit volume is unchanged.
Which shouldn't be surprising since the energy is stored in the dielectric by (e.g.) straining the molecular structure of the material.
The biggest reason for going to higher voltages is to reduce the interconnects, which get enormous at low voltages and high currents. (Cross-sectional area goes up inversely with the square of voltage for any acceptable IR loss, which is why long-distance power lines run at scary voltages.)