Not my understanding at all! The relationship that requires double the power when impedance is halved applies to all amplifiers of all classes.
Solid state amplifiers exhibit an anomaly at the crossover point from +ve to -ve output voltage. Class A amplifiers avoid this anomaly by keeping entirely on one side of it, let’s say the +ve side. To do this, the input voltage is DC offset (biased) to ensure the output never goes -ve.
There are a couple of consequences - a lot of output power is wasted as heat, and the input power consumed is constant whatever the audio output power. My Krell consumes about 700-Watts all the time and has huge fins to get rid of the excess heat.
A class AB amplifier on the other hand does allow the output to go -ve, at a point determined by the bias voltage applied. The idea is that the cross-over anomaly occurs when there is some musical content, and won’t be as obvious as when playing near silence.
Your amplifier manufacturer may be saying that his class AB amplifiers go into the crossover zone at much lower power when feeding 4-Ohms, compared with 8-Ohms. In this case, class A power would be defined as the most power the amplifier can deliver before reaching the crossover zone.
Seems to me that his AB amplifiers do not really have enough current delivery capacity in reserve, at least on paper! It would be interesting to ask what AB power they can deliver into 8-Ohms, and into 4-Ohms. Ideally it would be double, even if the pure class A numbers take a nosedive