Why does 60 Hz mean 60 refreshes and not 120?
Hertz means '(something) per second'.
With the light bulb, that's cycles of the mains supply frequency. Being a balanced a.c. waveform, that crosses 0 V twice per cycle. That causes the 100 light bulb flashes per second from the 50 Hz supply, or 120 flashes from a 60 Hz supply.
With the monitor, a cycle is a top-to-bottom refresh of the display pixels on the phosphor or LCD screen. There's no a.c. aspect in play here. If anything, the term could be applied to the frequency of a monitor's VSYNC signal, if its got one, but that's not why the term arose.
The unit 'Hz', originally used for a.c. waveforms, was the one later used for refresh rates in television and thence in monitors. The use of Hz was to do with how early television sets derived timing from the mains frequency.