Exactly!
This “color” in cheap devices they call it distortion, but in expensive devices it is “sound signature”
Yep, and what's more funny, you can realize that when you have really transparent setup - there will be great sounding tracks and awful ones. Once your setup has some kind of "sound signature" or "specific flavour" you realize, that all kind of tracks sounds very similar, no matter what album, or genre. This is very boring after a while and you start looking for "a change". Then it goes again and again in an infinite loop, forcing you to lose more and more money - now let's think about it
Once you have transparent audio chain you are surprised every time you listen to some new album (with timbre of instruments, dynamics, tonality, transient response, resolution, linearity). Sometimes it can be harsh, other time can be butter smooth but it can't be butter smooth all the time, even on harsh recordings ffs (not even mentioning losing a lot of details, resolution and speed on many of those "butter smooth tuned systems").
In my book that is the real term to describe transparency and HI-FI audio equipment. And guess what ? It doesn't always have to cost a fortune, you just need to be patient when looking for, listen carefully with your own ears and not some biased/sponsored reviews or let advertisment/PR do their job. You can also use information from this great site, at least to do some pre-selection of units that you may like, and at the end of the day, having not only something that "sounds subjectively nice to your ears" but also very well engineered product that you paid for.
I still believe there might be some kind of tests that we miss and that can show differences between dacs or amps, even though current tests say objectively there can't be a difference. Real life sound is never like 1kHz sin wave with certain amplitude. There can be some other phenomenons when complex waveform is being processed and a lot of interactions between them, which may affect final sound after conversion. For instance for poorly designed but great measuring headphone amp, thanks to high amount of NFB applied, there can be high level of TIM distortion (discovered not so long ago btw, in 70's if I remember correctly), which will never show up with sin wave 1kHz test etc. Even small amount of these can cause sound being always bright, tiring and unpleasant (thus I love multitone test and IMD which are quite important IMHO).
From the other site - TIM distortions usually rise odd order harmonics, so it would show up in THD test as sky high level of 3rd, 5th harmonics etc but I hope you get the point.