I wasn't born with perfect pitch but, over the past 45 years of constantly playing instruments - particularly guitar - I can hear a chord sequence or melody in my head which, when I pick up the instrument and play it out loud, is in the same actual key on the instrument. It's certainly not "perfect" pitch - which sounds like a curse to me in some respects :-) - but just the result of many, many years' playing.

Perfect pitch, as far as note naming goes, is of course a convention, rather than an absolute. It would be interesting to see and hear how people with perfect pitch - but with no musical education, training or knowledge whatsoever - responded to playing the same tune modulating into different keys. The convention for "Happy Birthday", when playing it at a party, dance, etc., is - in the UK at least - to play it in G. But that's merely a convention - Ab would do me just as well!