Possible, but I'm not even sure "tube distortion" is actually audible, as it has good chances of being masked; do you have measurements that correspond to your type of equipment?
Some tube amplifiers can potentially add 1% harmonic distortion, which is potentially going to be audible if you're listening to pure tones on some music. As you've mentioned, auditory masking effects will probably jump into play to make even quite high levels of euphonic distortion less discernable by the listener.
Of course, many tube amplifiers have quite low values of damping factor. This can result in large frequency response swings at the outputs of the amplifier when driving typical passive loudspeaker systems. These swings can be up to ±1.0dB, although probably ±0.5dB is more the norm. Such large frequency response changes are adding large amounts of distortion to the reproduced signal, noting that a 1.0dB change is approximately 10% distortion of the signal. If you equalised pink noise to match the amplifier's frequency response, and then compared it to unequalised pink noise, then you are highly likely to be able to hear the changes. Here is an example of the possible frequency response variations associated with amplifiers of different damping factors.
Looking at where those peaks and dips are occurring, it isn't difficult to take a guess at the euphonic colorations that are being added. Working through the frequency range, from left to right, let us see what we can hear. Such things as "the bass has so much more depth and punch", "the boxiness on the vocals is removed", "the extra clarity on female vocals makes them sound so real", "the smoothness of strings is enhanced to bring out their liquid nature", and "the edginess of horns and cymbals has been tamed". Of course, change over to an amplifier with a damping factor of 200 or more and all those superlatives just go and vanish into thin air, as we now have much more accurate reproduction.
Nevertheless, euphonic or not, these are simple colorations that anyone could take advantage of by simply purchasing a multi-band parametric equaliser, or even a simple third-octave equaliser. The latter were common enough in the past. That way you can choose much of the flavour of whatever tube amplifier you want at any point in time. And, one major benefit is that you can switch off these special sonic effects when you no longer desire them to be present.
