Electronic – How is the switch current related to output current in a DC-DC converter

Yes, Vicor, for example, makes a line of such converters, typically used for point-of-load conversion from a regulated high voltage (e.g., a 48V distribution bus) to a low-voltage, high-current supply for a CPU. (Note the option for "fixed ratio" under "Output Voltage".)

You could easily construct a boost converter that has the same property. In fact, any continuous conduction mode switched converter (buck or boost) operating with a fixed duty cycle has this property.

But such a converter would not be particularly useful in terms of using a 12V battery charger to charge a 24V battery pack. The converter would prevent the charger from getting any feedback about the state of the battery, leading to overcharging.

You need to be driving the MOSFET gate much better than what you appear to be doing to get it to switch efficiently. Look at the spec for the FQP30N06 - gate turn on threshold (Vgs(threshold)) might be as high as 4V. Also look at figure 1 - this confirms that you should be using a much higher gate drive voltage than what you are driving it with: -

Also, the edges of the waveform are really poor - you are not going to get decent rise and fall times from an IO pin because the gate-source capacitance is going to slow things down.

Consider changing your drive circuit.