The thing was, your post started off fine with the idea that IF X is audible, it makes sense it should be measurable.
Except throughout the post you kept implying that you ARE hearing differences and from that conviction...therefore it should be measurable.
There wasn't really an acknowledgement of the obvious variable that you could be misconceiving differences that aren't there. It was more about "I hear it...so why can't we measure it?"
"That is why I ask the ultimate question: If it doesn’t measure differently, does that mean ipso facto that there are no sonic differences?"
Remember that measurements, and measurement devices, didn't just arise out of some abstract vacuum.
The main reason measuring devices arose is to extend the known limits of our senses! That's why we need devices to detect X-rays that we can't see, telescopes to see distant objects we can't see, devices to detect radiation, ultrasonic noises we can't hear, and on and on. The measuring equipment used for audio gear can reliably detect and quantify both things we can hear, and cannot hear.
And the the use of measurements in audio only arose by correlating those measurements to what we hear. That's the whole point. It seems lots of audiophiles start with this strange assumption that measurements in regards to audio equipment are just some laboratory abstraction, whereas they arose by careful correlation to what we hear (and can't hear).
So, as to your question: It will depend on the particular claim. We don't have Absolute Certainty about anything of course, so it's a matter of adjudicating the likelihood based on what seems to be known. So if you take an AC cable, measure the signal with an expensive cable vs cheap cable and there is no detectable difference, that strongly implies it's not changing the signal. And therefore it's more probable a bias effect is responsible for the listener thinking he hears a difference. It's not Absolute. Just the more probable explanation.
Someone may object and say "Ok, but what If I AM hearing a real sonic difference that you can't detect by those instruments?"
Ok...how could that be tested? You CAN also test that claim: do a blind test to remove the possibility of listener bias. If you can detect the difference reliably, then even if the measurements are the same, this DOES suggest there is something audibly detectable happening. You are vindicated!
But if you are going to reject BOTH any attempts at objective verification (measurements) and subjective verification-with-controls (blind tests)...then what is left? What way forward to you have to figuring out what is actually the case?
If the way forward is "always trust our perception" then that flies in the face of all we know about the fallibility of our perception. And it also leads to countless contradictions where one person perceives "no difference" and someone else "obvious difference," which tells you NOTHING therefore about what is actually happening, and it would validate literally every crazy claim anyone has ever conceived (because no matter how crazy the idea, there are people who believe they are experiencing it).