Pro Micro - Measuring voltage between two pins, without using GND
No, you cannot - the ATmega's ADC is referenced to its ground, and there are also some strict limits on providing inputs more than a little bit outside of the range between its power and ground.

You may be able to read two analog inputs and subtract them for a sort of "software differential" analog input, but the signal on an LED is unlikely to be meaningfully "analog" anyway - it might be pulsed with a duty cycle, but it will probably not have a steady reduced voltage.

It's actually quite likely that the anode of the LED is connected to the toy's positive supply, and it is the negative path to the cathode which is being switched - this is the most traditional way of doing things, because historically semiconductor switches suitable for "low side switching" have had better performance than those suited for "high side switching".  There will probably be a resistor on one side or the other of the LED.  

With care and verification of suitable voltage ranges, you may be able to read the voltage on the LED cathode as a digital signal and invert it in software.