How is land subsidence measured? When groundwater is pumped out of the ground by wells, and not replenished fast enough, the groundwater level sinks. 

With it, the surface of the area sinks also. Locally, this effect is not apparent, because heights of objects are specified relative to the surface. The change in height stays zero by definition. But relative to the average elevation of surrounding land not connected in terms of ground water, there is a change. What changes is the distance between soil surface and center of gravity of the earth. The subsidence is a change in the shape of the earth, so its center of gravity changes itself, but the magnitude of that change is small enough compared to the surface change to ignore it. 

How can the change in surface elevation of a fixed point on the surface of the Earth be measured?

One approach would be to use the fact that the surface of a water volume in equilibrium is the same height everywhere, even if the surface is separated. The water level in both ends of a U-shaped pipe is the same. One could use a long pipe that reaches from the measured location to a reference point. Connected bodies of groundwater are large, so that may require pipes of a length in the order of kilometers. Also, with a large distance between the end the air pressure may be different, so there is no equilibrium. 

The ground water level itself is a possible reference point, because the ground water can permeate the ground. But we know that it is not in equilibrium, because ground water is removed only in parts of the surface - the wells, and the permeation is slow.

Ground elevation can be measured from satellites, as GPS does. I suspect that the accuracy is too low for many purposes because the relative change in the measured distance is very small. Also, the orbit of a satellite changes with the change of the center of gravity. Because the change is slow, elevation changes in distant regions influencing the satellite create noise in the measurement.
Land that has been heavily disturbed by humans in most developed countries, such as urban and rural regions will have been surveyed for topographical &amp; sub-division purposes.

Such data allows for a digital terrain model (DTM) to be created for such regions. When subsidence occurs, the subsided area can be resurveyed &amp; compared to the original DTM to ascertain the amount of subsidence that has occurred.

Undisturbed or poorly surveyed regions become problematic. The subsidence area &amp; the surrounding area can be surveyed, but then someone has to make a judgement call &amp; decide what the original surface may have looked like to infer an estimate for the degree of subsidence.

The other way is to have satellites periodically measure surface elevations &amp; then to do comparisons between each satellite measurement.

Radar could be used as the measuring system, but LIDAR (&amp; this reference also) can produce more accurate results.