The question is: If you can remember what C1 or whatever sounds like and then deduce from that, is it not just a way of more learning to be able to remember the other ones too?
Is there really a fundamental, functional difference between absolute pitch and "normal" hearing?
Yes, but that will take a long time and much dedication, and I don't know anybody interested in trying to remember many notes. There is also the risk that the more notes you try to memorize, the less reliable it gets. It's a lot more practical to just remember a few anchor notes and put the effort into training relative pitch.
For the purpose of performing music (or any practical activity), I'd say a good relatively pitch and a good memory of a few anchor notes, is as useful as absolute pitch. Functionally, AP isn't much more valuable.
But I do think, based on observation, that there is something fundamentally different with AP. It's like they hear sounds very differently, and their brain has a special mapping of frequency that I don't have.
They recognize very subtle deviations in frequency. I have had a person tell me "It's too sharp" when I was only off by 3 cents according to the tuner.
They don't make mistakes and they are fast, almost naming the pitch as soon as they hear it. It's effortless, precise, and accurate.
Ok, but THAT I have. I don't have absolute pitch, but I was always able to tune my cello more precisely than my teacher. (and other instruments too, not limited to one person)
Another trick: tune then relative to the first string tuned, not incrementally from one tuned string to the next. That way you end up much closer to the ideal.
To give an example. I was playing board games with an acquaintance with AP. Someone tinked a glass or something (brief, very high pitch / edge of musical range). She named the pitch without hesitation, and noted that it was slightly sharp. (I of course couldn't confirm, but had no reason to doubt her.)
I have very good relative pitch and pitch recall. E.g. I can tune a guitar quickly by ear without relying on beat notes, and can do so without a prior absolute reference (I have memorized E, A, and Bb), though I may be ~30 cents off the reference. I can reliably identify the key of a song I heard months or years ago by comparing it to memorized references.
But I have to think about it, imagining and computing pitches and intervals in my head. And hearing other pitches in the interim throws off my recall.
To do these things instantly, with precision of cents, is magic to me. It's the difference between a savant producing the product of two large numbers instantly, and me working through the multiplication algorithm in my head.