It looks like an electromagnet.
Start with a DMM - check the coil winding resistance. From there, you can probably make some educated guesses about where to start testing using a benchtop DC supply. Slowly increase the supply voltage. Keep tabs on when it starts to get hot, and you've likely found its continuous current limit.
While I've never heard of this circuit before, I watched the video @user23711 linked. I think I can give you a decent explanation.
Essentially, you have an AC source driving your device with a resistor in series acting as a current limiter. The voltage measurement is simple. You are literally just measuring the voltage across the device under test, just like any other voltage measurement with an oscilloscope or multimeter.
As for current, you are measuring the voltage across the resistor and not the device under test (DUT) because voltage is (sort of) proportional to current in a resistor. Since the resistor is in series with the DUT, the current is the same as that of the DUT. However, the voltage across the resistor will be proportional to the current while the voltage of the DUT has an unknown relationship to current. That is, after all, the purpose of this circuit.
Now, if you had a current probe for your scope, you wouldn't need this test setup. Also, be mindful that the V/I curve for a resistor is not perfectly linear. If you are looking for high accuracy this might not be the way to go.
Finally, the reason for the AC source is to frequently enough cycle the voltage of the DUT. Then, when using XY mode on the scope, the IV curve of the DUT can be seen. But really, you could use any source you want (that won't break the DUT), and view the voltage and current curves with respect to time.
Best Answer
Schneider are being a bit coy about exactly what's going on inside the transmitter. You have linked to the CE certificate which isn't much use. The Wireless and batteryless pushbuttons Harmony XB5R (2017) is more relevant.
Figure 1. The system appears to be very flexible with one button able to actuate multiple receivers or vice versa. This opens up all sorts of possibilities for use on forklifts, trucks, etc., for interaction with fixed installations.
If, as we seem to all suppose, they are piezo generators then an introductory article by American Piezo may throw some light on the subject:
They also have their APC piezo calculator which you may enjoy. I haven't made sense of it yet.