Why is the puddle underneath a melting ice cube as cold as the ice cube itself

It's not, if it were it would be ice. The water is a bit warmer.

Because the electrons in a solid, in this case the ice, are only vibrating. And so when it heats up, the electrons vibrate more. Imagine a scale from 1 to 150, 1 to 50 is a solid, 50 to 100 is a liquid, and 100 to 150 is a gas. If the ice has just been melted, then the electrons are only just in the beginning stages of moving enough to be a liquid, and the movement of electrons determines the temperature because it creates energy. I hope this was the answer you were looking for, let me know if I confused you