The short story is, I don't have it.
Absolute pitch, formerly called perfect pitch, is the ability to name a pitch after hearing it. It's to be distinguished from relative pitch, which is the ability to name intervals after hearing two notes in succession, or at the same time; or copy melodies etc. There's different kinds: some people can do it in some contexts and not others. It's said to be rare: possessing "perfect pitch" is often included in the litany of inborn genius characteristics of the young prodigies, "one in ten thousand". People are studying it, though, and it's a little more complicated than just being a God-given gift to the chosen few. Learning music when being young helps, speaking a tonal language helps. Nearly everyone has some aspect: eg if you ask musically untrained members of the general public to sing a currently popular song, they start in the right key much more often than you would expect from chance. I've known people who had it, in a useful sense: my junior high orchestra director, for instance, would hear an airplane flying overhead, and announce that it was an E, but somewhat flat. And my son's AP is pretty good. I've never had the sense of having perfect pitch, but I feel like I do have some aspects of it. I can imagine playing any note of the bassoon, for instance, hearing the pitch and the timbre in my head, and imagining what my fingers and air column are doing. I'd be surprised if the pitches I'm imagining were that far off. And once when I was trying to learn intervals, with my wife playing them at the piano, she noted that I was always getting major and minor wrong, but was quite accurately calling white notes major and black notes minor. It's wrong, sure, but your brain can't make that kind of mistake without knowing, in some sense, what the absolute pitches are.
So I was intrigued when I read about an online study of AP. Here it is, out of UCSF. They are interested in the genetics, so there are survey questions about family background, musical training, and languages, before they hit you with an online pitch recognition test. I found the pace pretty quick, you get maybe one second for each tone, and I didn't feel like I was doing any better than guessing. About halfway through it switched from pure sine wave tones to piano notes, and I felt like I was doing even worse. I didn't care, I just wanted to finish. The results reflect that: I scored 14.75 on pure tones, and a 3 on piano tones; this is out of a best possible of 36. The average score in their test-taking population is a touch over 17, and random guessing would give you 7.5. Their cutoff for having AP is 24.5. There's a plot of where I fit in: