What is the reason for multiple ground connection and multiple testpoints on this PCB?
You are on the right track with your comment about the "same potential."  PC board traces have non-zero impedance, but have resistance and (more importantly in this case) inductance.  This means that when you are running a load with potentially high, switched currents, the instantaneous potential is not the same at all points along the current path.

Your board has established multiple current paths; some are low-current and these will be pretty close to the same potential because there is little voltage drop associated with the current.  The ground for the circuitry on this branch will be the more or less the same between devices, making this ground branch useful for a reference for analog signals and for digital thresholds.

The path containing high, switched currents can have relatively large instantaneous voltage drops along its length; one volt or more is not uncommon at the switching edges.  Imagine the ground reference on a processor moving one volt with respect to the digital input signal levels and you can see how this would be a problem.

The strategy is therefore to design so that the high-frequency, high amplitude currents are not flowing in the sensitive, low power circuitry.  Similarly, to measure voltage on a device, the test point used for ground reference should be local to the device.  You can experiment with an oscilloscope, measuring a voltage at one point in the circuit while moving the probe ground to different points and you will see the effect.