"...Why is it necessary for solid soap to cure, while liquid soap is okay to use as soon as there is no zap?..."
I have to talk about solid soap first to answer your question, so here goes --
Any solid soap, including the Gent's softer shave soaps, need time for three things to happen -- to finish saponifying, to evaporate excess water, and to develop a crystal structure. Even if your recipe is designed to make a long lasting, mild, lathery bar, you will see the best of these qualities when you use a soap that is given enough time to fully cure.
The first aspect of cure is to complete the last lingering bits of saponification. When you unmold a soap, the saponification might be 99.9% done, but it's good to allow that last 0.1% time to finish before using the soap. Anyone who has eagerly bathed with a freshly cut bar of soap and gotten some skin irritation has learned that lesson. (... Of course I've NEVER done that!
...) A few days should be enough to meet this goal for a properly made soap.
The second aspect of evaporation is the most visible process to us humans. Evaporation causes a bar of soap to lose weight, shrink somewhat in size, increase in hardness, and become less soluble in water. Y'all can discuss this one to death without my help.
The third aspect -- development of crystalline structure -- is a process that most people can't see and so they tend to ignore or discount this as not being important.
The crystal structure of soap is made of soap molecules that tend to join up to form large sheets. (The soap molecules form other shapes as well such as balls and hot dogs, but please bear with my simplification.) Ideally, these sheets stack one on top of the other, just like slices of bread in a triple-decker sandwich.
In this ideal structure, the spaces between the sheets are filled with a alkaline liquid made of water, glycerin, dissolved soap, etc. This liquid is much like the filling in my sandwich. Some of the alkalinity of this liquid is a natural consequence of lye soap being lye soap, and some may be bits of excess lye that will be saponified and neutralized during cure as mentioned above.
When a handcrafted soap is newly made, the soap structure is disorganized (small wonder after being wildly beaten up by a stick blender!) and there is a lot of free liquid floating around. This disorganized situation is a bit like a dry sponge lying in a puddle of water. You want the sponge to soak up the water to clean up the mess, but the process isn't instantaneous -- the sponge first has to get damp and then it can absorb the water. Same with soap -- the alkaline liquid needs to be "soaked up" and trapped within the soap structure, but it takes time to create an organized structure that can soak up the liquid.
A crystal structure of sorts is developed in commercial soaps by drying soap flakes under vacuum, extruding the flakes through dies, forcing the soap into a mold under pressure, and sometimes by (French) milling. Commercial soaps are also sometimes also given a certain amount of time to quietly cure on their own.
Since we don't normally have the ability to artificially dry, extrude, mold, or mill our handcrafted soaps, our soaps can only be cured by the application of time.
So on to liquid soap --
As Susie pointed out, liquid soap doesn't have a crystal structure, and it doesn't need to dry. A "cure" time for liquid soap is mainly to allow time for the last bits of saponification to finish up and allow any particles that might be in the soap to sequester (settle out).
I have to disagree about a liquid soap not being harsh -- it all depends on the fats used to make the soap and the amount of dilution to get the final product. Since liquid soap can be made of any fats, including coconut oil, it can be as stripping and harsh as any bar soap -- perhaps more so since the superfat in liquid soap is usually low. Dilution can modify harshness -- more water in the diluted soap will reduce the cleansing action.