Is there a dual of the transformer?
Actually, this is a common thing to wonder.

There is a dual to this. When you have devices sharing a common winding and magnetic flux(Magnetic "current") it makes a perfect dual for devices sharing a common electric conductor. Nice picture from wikipedia:

2

You can also take a look at "Magnetic Circuits". You can start learning some fun terms when you go into these concepts in detail, like "Magnetic Capacitance," it seems that my flux does have capacitance.

The way you can determine how much energy passes through a transformer can be broken down into a magnetic circuit that works just like an electric circuit with different units. Magnetic Circuits are an analog of Electric Circuits, which are much easier to work with for many many reasons.

Think of it like a voltage source or a current source. They are direct analogs, but when you build a voltage source it is a hell of a lot easier than a current source.

Side Note

The magnetic flux is shared in a core due to the fact that magnetic flux is perpendicular to the wire, the issue with electric flux is that it points between two surfaces, not looped around. If it looped around a dielectric would get the job done.

In relation to the capacitor inside the other

IF the smaller becomes bigger, it will end up acting like two coupling capacitors with a series resistor between them, as it gets smaller, the overall electric field will be minimal, but you could put a big big E-Field in there, not nearly as effective as a transformer.