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

The answer is that the system is in 'equilibrium'. More exactly thermal equilibrium. Ice and Water can both form at 0C and 1 ATM. When two things are touching, they are at equilibrium with each other. Since the ice cube has had enough time to melt, it has had enough time to reach equilibrium with the water and is at 0C.