No. Either should have adequate damping and not audibly alter the sound.The only things that will affect the "sound quality" of an amplifier are power, distortion, noise, gain & frequency response. If all of those are within inaudible tolerances, you won't hear a difference.
I "almost" agree with you here: most modern amps are good enought that under certain conditions sound "almost" identical. Other things that used to affect amps sound (like slew rate) have improved so much with modern components that even a cheap bazooka amp will have more than acceptable values...
However, why do many of us hear differences between ZAPCO 360.4 and a Sony Class D? Are we all delusional and should be buying the Sonys instead of waisting cash in "high end" amps? Yes and No.
Yes: getting a reasonably good amp & spending more on other parts of the system (Install, Drivers, damping, etc..) will likely yield higher gains PER DOLLAR.
BUT (and this is a bigger butt than Kim Kardashian's) there are sonic differences. Why? Because the tests that you mention only cover some of the variables that play into how an amp works in the real world. We do not always measure some of the things that can have a big influence on the sound.
Push the mentioned Sony amp to a level that the transients start clipping and you will be able to tell the difference from the Zapco.
Just for the record, these are some things that can also affect how and amp sounds:
Output current (not power), output impedance, open loop bandwidth, transient intermodular distortion, phase response, feedback ratio...and keep counting.
So if someone says that a class D amp sounds differently than a similarly rated class A/B there is a good chance that there is indeed something different.
My wife can tell that full range class D amps sound a little funky in the top. I cannot. At the Boston Children's Museum there is a station that measures your frequency hearing....mine started fading at 19 khz, she could still hear above 20khz. I could hear as low as 20 but she couldn't pickup anything until 22hz...that could be where the difference is.