Amp Measurements

The measurements you point to have VASTLY different scales than those of headphones.
So at a first glance they appear to differ considerably but these scales are chosen to make the very small differences visible.

With headphones the FR scale runs from 20Hz to 22khz, maybe sometimes from 10Hz to 24kHz.
The vertical scale (from top to bottom) may be between 30dB to even 60dB where the divisions are generally 5dB or 10dB

In case of the amplifiers the FR runs from 10Hz to 100kHz and the vertical scale is just 6dB from top to bottom with 1dB divisions.

When you observe these scales and 'stop' at 20kHz reading those scales you will find that only 1 amplifier rolls off (and on purpose I may add !) just 1dB at 20kHz and that is the Meier Corda. I know that a roll-off of -0.5dB at 20kHz can still be detected and sounds 'fuller' not rolled off for some reason.

headphones can vary 10dB to even 20dB between 20Hz and 20kHz and may even exhibit dips of over -30dB while amplifiers vary less than 1dB over that range.
Perhaps with the exception of the lows with some amps.
Note that roll-off is only present at the extremes and all amps (but one) are flat within 0.1dB from 100Hz to 10kHz.

When you overlay these with headphone plots you will see no differences at all in FR (when the output R < a few Ohm).

I know Purrin has measured 1 headphone with 2 different amps (I thought it was the Magni versus O2) and looked for the plots but can't find them that quick.
The headphones measured exactly the same !
That is including distortion and FFT measurements.
I have done this myself a couple of times and found the same.

The reason for that is very simple... the distortion products a headphone 'creates' (the harmonics) are tens to thousands of times higher than those the amplifiers make and thus are masked by the signals the headphone creates.
It doesn't mean the differences aren't there but they added signals are so small you don't even see them on those plots.

THD figures and plots differ between all amps and even quite a lot.

I should write an article on this as to 'asses' performance of an amplifier based on Tyll's (and others) measurements you again have to take all the different plots into account to be able to say something about audibility and the matter is VERY complex.
Looking at the FR alone tells everything about tonal balance but the added harmonics ALSO have an influence of tonal aspects but need to be significant to do that. Peaks of -100dB in an FFT aren't significant in any case.

Too complicated to put in a post though because of the many aspects involved in creating 'the numbers' and plots and the conditions these plots are created in.

Yes, they are mostly consistent but of course there are differences that matter and other differences are lower in amplitude than we can hear on certain volume levels BUT may just become audible at higher volumes.
This too is a dangerous path as when playing louder we get close or OVER the limit an amp has (clipping) and in this case audible differences at higher levels do not come from the amps design/assets but are simply caused by lack of headroom/clipping.
Hard to tell unless you are measuring the peaks.

The biggest differences are caused by the output resistance, differences in power (voltage) levels by different gain/volpot settings, and the distribution AND levels of added harmonics.
Let's not forget time of day, mental state, preferences, biases.