If you apply a constant current to a capacitor, the voltage across the capacitor will increase linearly as the capacitor stores energy in the electric field.

If you apply a constant voltage to an inductor, the current through the inductor will increase linearly as the inductor stores energy in the magnetic field.

Perhaps part of why the intuition can break down is that in real life, inductors tend to be much "leakier" energy storage devices than capacitors. If you store some energy in an inductor and then change the voltage across it to zero (practically: short its terminals together), in theory a perfect inductor will maintain a constant current forever and the energy stored does not change. In practice inductors (with an exception for things like superconducting magnets) are made from wire that has a resistance, and so the current in a real shorted inductor will eventually decay to zero. This means that in practical terms inductors are mostly only useful for short term energy storage. On the other hand, real-life insulating materials (like air, vacuum, or Teflon) can can be pretty close to perfect insulators allowing real capacitors to store energy more or less indefinitely... certainly on timescales of years.