I propose that it's not that we can't measure things that some can hear, but we've not yet learned "how" to quantify/measure/isolate everything that we can hear. We're getting closer and closer though. Our understanding of HRTF is pretty complete, for example, but we do not yet have total understanding of how someone's brain processes sound. I think also we can often overlook seemingly inconsequential things that many can hear, such as extremely minor deviations in frequency response, for example. Another thing to keep in mind is that we, as a species, are sometimes pretty good as comparators, but typically very poor in realizing absolutes, due to accommodation.
Can you add anything to the definition of "what is this thing that can be heard, but cannot be measured"?
Everyone, including myself, is talking about a "something." Let's try to define it. What is the "something"? If it can be heard, it can be described. If it is heard as the 'difference' between sounds, what is this difference?
I have never heard the thing that can't be measured so I can't describe it. What I can describe is something I "thought" I heard between two USB cables. What I thought I heard was a (slightly) clearer, brighter presentation from an old 1.1 ferrite beaded cable (from an old CD burner) over a new 2.0 cable. It certainly sounded like something that would be measurable if I had the equipment. There was a frequency response in the brightness that should be measureable and maybe a clarity in the bass that might show up on a spectrum analyzer as having less fuzziness around the fundamental notes.