How are the COP values so high? The spec on the Vaillant aroTHERM+ says ~2.9 but that is showing well over 5 on some; what's the difference?
A single COP is some yearly average usually but in these measurements it is probably a shorter period.
In europe there is a somewhat more rigorous measure of Seasonal COP (SCOP) where the typical climate (in my case nordic) is taken as a standardized test. So x number of days outside temp 0C, y -5C, z -10C, w +10C etc.
So in case of Air source heat pump if the outside temperature is +10 or +15 you can easily get COP numbers that are 9 or above.
Better technical documentation usually has rated COP for some different outside temperatures or even a graph.