The gist of it is that a combination of measurements and well conducting blind listening tests suggest that once certain thresholds of frequency response, distortion, and such are reached the human ear can no longer hear what the component (amp, DAC, whatever) adds to the sound and the component becomes transparent. Keeping things like current and voltage output constant any piece of transparent equipment should be indistinguishable from another piece of transparent equipment.
Things do get a little fuzzy around the edges. Although the listing tests suggest that such limits to human perception do exist and are achievable, research has not been comprehensive enough to determine exactly where those limits are. Beyond that, there probably are a very few legitimate "golden ears" out there who can hear things that others can't but unless you know that you're one of them you probably don't have to worry too much about overkill measurements in the same way a normal jogger doesn't worry about the custom fitted running shoes that an Olympic sprinter needs to shave a fraction of a second off their time.
There are other people with differing opinions. Some people just prefer a certain coloration from their equipment. Some may admit it and others may not. Maybe a couple really are golden ears and really can hear things below the usual threshold and dismiss what we know about most people's hearing because they're special. Some dismiss the concept of measurement and blind testing outright. They insist that their ears are more sensitive than any scientific instrument and they declare themselves immune to well known cognitive biases. Many are somewhere in between and don't want to dismiss either their subjective experiences or the objective measurements.
I pretty much go with the measurements and blind testing. They aren't perfect, but if your goal is accuracy to original signal it the best we've got so far. I've got nothing against liking a particular coloration but I think its much easier to reach such a goal by removing as many variables as possible by going with a transparent DAC and amp. After that pick the headphones you want and then fine tune them with either mods or EQ and other DSPs.