What is the difference between a conventional capacitor and a supercapacitor?
In an electrolytic capacitor, the capacitance is in the insulating material between the electrolyte and the metal, normally a metallic oxide.
A supercap is a kind of electric cell, a battery. It works by ionic migration and redox reactions. The amount of migration required is just electrical alignment, and the redox reactions are just with adjacent molecules, so speed is fast (for a battery) and charge storage is low (for a battery).
A double-layer super-cap has a bound charge layer protecting one electrode, so the battery is in series with a single-molecule-thick capacitor. This very thin capacitor has very high capacitance, because it is so thin.
The liquid electrolyte doesn't have to be liquid by any other measure at all: it just has to be liquid enough to allow the formation of the bound charge layer, possibly rotational ionic migration, and possibly adjacent redox reactions.
The picture you linked to, showing electrodes separated by a electrolyte and separator, does not show a solid dielectric. They've perhaps had a brain freeze and misunderstood the mechanical separation of the two electrodes as being something else.