When I was working in the Music Tech department in college, I had set up a mic and frequency meter for a lecture/demo. A number of our staff were long practiced with tape based kit and had spent many years having to use a 1000 Hz tone to set up tape machines.
It was interesting to watch each one look at the rig, walk up to it and attempt to whistle a one kilohertz note.
They were all , including me, able to hit it within a few cents- that's a hundredth of a semitone.
It is clear that a pitch that is very familiar can be memorised accurately. I suspect that you can remember the Oboe "A" that an orchestra uses as a pitch reference too. A feel for intervals would allow you to pitch other notes too- though I very much doubt that this is how it's done!
The problem for the sufferers from 'perfect pitch' is that they care about it. Just because a note is numerically perfect in frequency, does not mean it's the note I want to hear.
To me, a singer or player that has perfect, unwavering, nail-on-the-head pitching, and metronomic timing sounds souless and inhuman- or more likely these days, to have been heavily processed in post production.
Cheers
Dave