This is the false statement, and is the typical statement of amp manufacturers that can’t double down with their amps.This statement is false.
I recently serviced out a Realistic receiver with 18 watts. Even though it has no chance of doubling its power at full power, at power levels less than full output, it does so with ease, for the simple reason that despite its inadequate supplies and heatsinks, the amp behaves as a voltage source. Its only that last 3dB into half the impedance that is being touted as such a great thing- and really has very little effect if any on how the amp actually sounds.
Any conversation otherwise really points to not understanding how a voltage source works, and why its important. Now its been pointed out that no amp can really double its power at full output when the load is halved. This is true because of a thing called Ohm's Law. It has to do with the output impedance of the amplifier being finitely more than zero. Whatever that value is, it will always prevent the voltage amp from perfectly doubling power as the load impedance is halved ***at full power***.
Since actually doubling power at full power can't be done, how important is it? If sound quality is your goal, its likely that the amplifier topology will play a bigger factor in the sound of the amp, on account of how it makes distortion, and how audible that distortion is, especially at lower power levels where the amp spends most of its time with real music signals. Its also unimportant because you can't operate the amp constantly at or near full power due to the dynamic nature of music.