Electrolytic capacitors are closer to batteries than to non-polarized capacitors. Lithium-ion cells in capacitor housings also exist, presumably to evade tariffs and restrictions involved in shipping batteries.
Electrolytic capacitors use the chemistry to make a very high dielectric allowing the plates to get very close and increase the capacitance and decrease the size.
A cell in a battery is a capacitor then converts the charge on the plates into chemical energy and vice versa allowing much more energy storage and a flat operating range as the plates charge is replenished by the chemical reaction.
This article doesn't go into details but it sounds like the breakthrough is a much better dialectic then storing energy in a chemical reaction.