Can a water vortex (whirlpool) be used to efficiently store energy?

Yes, this would be horribly inefficient compared to pumped hydro or even a regular flywheel.

With a rotating fluid, there's a lot of viscosity. This viscosity generates heat and slows the fluid down. You would be able to offset this somewhat if you kept the container for the fluid moving with the fluid itself; but even then I believe there would still be significant losses from internal viscous effects. You would then also just have what is essentially a fluid filled flywheel, at which point you have to ask yourself why you didn't go with a solid material of higher density and likely better strength/stability.

The advantage of pumped hydro over this vortex system would be because a pumped hydro system doesn't require constant motion. It stores the energy as potential, so it only loses energy to viscosity when it is moving the fluid to the storage tanks. The less you have to move the fluid, the less you will convert energy to heat through viscosity.

It's an interesting idea, but I don't think it would be practical.

As mentioned by other writers, water viscosity will rapidly convert any kinetic energy into heat. However, efficient energy storage could be attainable with superfluids. Superfluids are related to Bose-Einstein condensates. (but they are not always interchangeable) Their special property is having absolutely no viscosity, so a whirlpool of superfluid will rotate indefinitely. The primary issue with this is the cryogenic temperatures required. Helium-4, for example, requires temperatures below 2.17K to reach superfluidity. The energy required to create and maintain such temperatures could exceed the amount of energy stored, making the system useless. Harvesting the energy could prove problematic. Since there isn't viscosity, a turbine would experience no drag. (to be precise, very little drag from residual viscosity)