Absolute Pitch research, ear training and more
Naming notes is merely the most obvious consequence of absolute pitch ability. If someone taught you note-naming and said "You now have absolute pitch!" I consider that equivalent to teaching you to recognize and say all the Chinese letters and telling you "You now know Chinese!" The ability to name and recall notes is an interesting party trick, but musically irrelevant.
There is no scientific evidence anywhere of any adult ever learning perfect pitch by any training method. Historically, it has been shown that these methods can indeed improve pitch recognition and recall, but the acquired ability does not resemble "natural" perfect pitch, and follow-up studies have shown that the ability disappears without daily practice.
With the exception of pure sine waves, any pitch that you hear is going to be accompanied by lots of extra information provided by the device or instrument that you're using to make it. Whether you carry a pitch pipe around with you or come back to the keyboard every two hours to "check your memory", your mind is not hearing a pitch but a complex waveform.
It's possible that you could, with great effort, remember the entire experience of this complex waveform, but what you will have learned is just one note on one instrument, not the universal character of a pitch. Furthermore, listening to pure sine waves does not eliminate the effects of timbre; a quote from Harmonic Experience explains why.
..In the world of audible sound the naked sine wave is denied us. Why? Because waves make waves. A pebble dropped in a puddle makes ever smaller, ever finer waves. The same is true within the chambers and fluids of the ear. The very act of hearing even a simple thing involves complex waves-within-waves.
The primary flaw in this notion is that pitch recognition is listening. When you pay attention to how your head feels, you are focusing your attention in the wrong place. Would you recognize the taste of butter by analyzing how the toast crunches? Would you recognize silk because it was shaped like a tie? Trying to remember how your head feels when you sing a pitch is simply the wrong thing to be attending to.
You will associate extra feelings with the sensation of "pitch" which are not universally attendant. Your voice changes. It changes not merely as you age, but based on how you feel on any given day. If you have the flu, your throat will not feel the same as it does as when you are healthy. When you've just woken up, your throat is more relaxed than it is after a stressful afternoon at the office or an exhausting game of volleyball. The "feeling" you are attempting to associate with pitch fluctuates unreliably.
Using your voice is still an essential part of learning perfect pitch; but you must pay attention to the sounds you are making, not to the feeling of your musculature.
Although the fact that non-musicians have AP could be (and probably is) evidence that AP occurs spontaneously, the ability to perceive pitches is completely separate from the ability to play or compose music. You can know all your colors without being an artist or a painter; likewise, you can know all your pitches without being a musician.