This is easily provable as not true, just by looking at the measured test results on Audioholics. The signal-to-ratio measurements and the FFT distortion analysis will usually show clear differences. The noise floor in particular can vary by at least six decibels, sometimes more.
I will agree that, at least for high quality dedicated power amplifiers, amplifiers are becoming more alike distortion-wise. I believe this is due to the increased use of IC op-amps in the two or three initial amplification stages, as opposed to using custom discrete-component circuitry. SNR measurements still seem to vary quite a bit, though there doesn't seem to be a pattern to it. Some expensive amplifiers are noisy and some AVRs are quiet.
On this forum there are some contributors who believe all solid-state amplifiers are audibly impossible to differentiate between, and others (like me, TLSGuy, and Gene, to name three who have posted statements about it) think otherwise under some circumstances. My conclusion has been the smartest and most reasonable posters stay away from strong assertions and absolutes.