Depends on the level you’re viewing it from.

At a high level, yes, the purpose is basically the same as a gas tank. You put something in (electricity), and take that something back out again later. Though it differs from simple physical storage in that you get slightly less back out than what you stored.

At the level of the actual physics of the component, no. Gas goes into a tank as gas, remains gas in the tank, then come back out as (the same amount of) gas. Electricity goes into a battery, is converted into chemical potential by using that electricity to move ions from one side of the battery to the other, and comes back out by reversing that reaction. There is no electricity in a battery that isn’t charging or discharging.

If instead of a chemical battery it was a mechanical spring that was wound up with an electric motor, and you extracted the power by running it in reverse as a generator from the energy of the spring, is that still storing electricity? A battery is the same, just chemical instead of physical.

Except a battery is converting chemicals and using their reactions to generate electricity. Think of it more like a reservoir with a hydro plant. Your feed in electricity, the hydro plant pumps water in the tank, then during a blackout? The reservoir empties itself through the turbine, re-generating electricity. The cars battery does a similar thing. It takes electricity to change the state of the chemicals, then lets the chemicals react to generate electricity.