Amplifier circuits multiply impedances. For instance in a common emitter amplifier stage around a BJT (bipolar junction transistor), the input impedance is much larger than what is implied by the emitter resistor. In effect, the transistor multiplies the apparent resistance of that component. If a capacitor is put in parallel with the emitter resistor, that capacitor looks smaller: from the base current's point of view, the capacitor is filling up very rapidly (which is due to the collector current) making it look shallower. The feedback capacitance in the stage is multiplied though (e.g. capacitor connected from collector to base, or such a naturally present parasitic capacitance). This is called the Miller effect.

Though this op-amp circuit creates a virtual larger capacitor, it's not useful as a general capacitor. You can't use this to make a 10,000 uF cap to use as a power supply filter; it doesn't actually have backing energy storage to deliver the transient current on demand. Now, yes, if that op-amp itself has a huge power reservoir behind it, then maybe this small capacitor can be made to work like a huge one in this manner. But it will be the op-amp circuit and its supply doing the actual work. This is essentially a voltage regulator.

In a voltage regulator circuit you have an amplifier with some capacitors upstream of it. The amplifier amplifies a fixed DC reference voltage to the desired voltage level. It holds that level very stiff: much more stiffly than those upstream capacitors would be able to on their own, so effectively it multiplies them.

Another thing you can't do is use it as a coupling capacitor, which is very useful role, allowing an AC signal to pass between two domains that are biased to different voltage levels. In the circuit, one end of the capacitor is grounded, so ...

I speculate it acts like an inductor because that would be a logically silly thing for it to do but I'm not doing the math at 1am on my phone.

I've always admired lcamtuf.

No offense, but these are words typically spoken by someone who has never taken any electrical engineering courses or read any of Walter Jungs books, after having some electronics background.

"gotten really good"? Really, like in the 1970s?? (the 741, still widely used today was released in 1968). I recall in the 1980s in the electronics that were once completely discrete, they were _everywhere_ and still are today.