How both live and neutral wires can have such small voltage difference in relation to the ground in this faulty wiring?
The most likely explanation is that your "ground" isn't actually connected to anything; it's simply floating.  There may be a bit of stray capacitance between the ground wire and the live and neutral, if only because the wires are running together in the power cord.  The leakage through that capacitance will be enough to show as a small voltage in your multimeter.