From this question I've learned that after a propeller change the plane should go through a test flight. Why is this?
The purpose of the test flight is for the safety of passengers. For most maintenance, in a not-for-hire aircraft, a private pilot can perform the test flight and determine the aircraft is airworthy, but they must do it without passengers.
Often when people talk about propellers, they think of them as a solid piece of material attached to the crankshaft of an engine. This is true for small planes, but absolutely untrue for larger aircraft.
On small aircraft, you might do a test flight to insure that the propeller functions properly in all flight modes. Mostly you'd be looking to insure that replacing the prop actually fixed the problem your were having. Defects in a fixed prop might make it vibrate due to not being properly balanced, which will show up on runup, or it might have an internal flaw that will only be apparent at particular flight modes, like ascending, descending or level flight and turns.
On larger aircraft, propellers are more sophisticated.
A reciprocating engine tends to develop maximum power in the 2200 to 2500 RPM range, so the engines are set to run in this range and the actual thrust is modified by changing the angle of attack of the propeller blades. This is usually accomplished by either an electric motor or a hydraulic piston. The motor or piston is in the hub and geared to the blades.
This makes the overall propeller system much more complex, and requires exercising it before certifying it flight ready even more important.
Adjustable propeller blades can be turned to a negative angle to allow engine braking on landing, and backing up on the ground. It can also be 'feathered' if there is an engine failure, so that the engine will not continue to turn over if it is shut down in flight.
There are other adjustable propeller systems that use air pressure or mechanical gearing.
Gas turbine engines have an even narrower maximum power range, and require even more complex propeller systems. They turn at around 10,000 RPM and use a gearbox to turn the props in the 2200 rpm range.
During my time in the Air Force, I changed propellers on C-124s, C-130s, C-133s, and a few other propeller driven aircraft. None of these changes required an actual flight verification, but did require a runup with a checklist for specific verification. Of course the pilots and flight engineer would be aware that the prop had been changed.