Energy stored in a magnetic field is extracted when you use it, and the field weakens. This is a bit of a simplification but close enough.
The energy in an inductance L is E = \$0.5I^2L\$. To store much energy that way relative to a battery you would need a lot of inductance and/or a lot of current. All superconductors have limits in current density and limits in magnetic field (and lower limits in combination) so you can only go so far depending on the material, temperature (colder is better) and whatever enormous pressures you can subject it to (more pressure is better).
It's not clear to me how you would extract the energy- usually energy in a superconducting magnet comes out as heat when the material goes normal (quench) and boils off a lot of cryogens quickly. That's not the most desirable form of energy. Getting it out as electricity without losing too much of it (as in the typical heat switch used in superconducting circuitry) might be tricky.