I've never given much thought to the subject; I can't think of any reason why it couldn't be learned, though I'm not sure a really expensive training course would be worth it.
I think when we name colors, our brains subconsciously compare the colors to colors we've seen before, so perhaps perfect pitch is a matter of memorizing pitches and comparing them to the ones you hear. (Is that even really perfect pitch? Isn't it just relative pitch with some of the pitches being ingrained in the brain?)
If a person with perfect pitch can name what notes are what, doesn't that imply he's learned the names of the notes? And doesn't that imply he already has a mental concept of the sound of those notes?
Piano keys are tuned at certain intervals, but in the real world sound frequencies wouldn't fall on exactly one pitch. It would be interesting to test people with perfect pitch (who have learned the names of the notes) to see how precisely they can recognize "in-between" frequencies, and how precisely they can recognize "mixed-up" frequencies (a bunch of different frequencies at once). These experiments have probably already been done, but I'm too lazy to try to look for them right now.
In his book This Is Your Brain on Music (a very interesting book), Daniel J. Levitin also brings up colors:
Most AP [perfect pitch] possessors can name the pitch of other sounds, too, like car horns, the hum of fluorescent lights, and knives clinking against dinner plates. As we saw earlier, color is a psychological fiction--it doesn't exist in the world, but our brains impose a categorical structure, such as broad swatches of red or blue, on the undimensional continuum of frequency of light waves. Pitch is also a psychophysical fiction, the consequence of out brains' imposing a structure on the unidimensional continuum of frequency of the sound waves. We can instantly name a color just by looking at it. Why can't we name sounds just by listening to them?[I don't think this means color and pitch perception are necessarily "linked" ... they are just similar.]
Well, most of us can identify sounds as effortlessly as we identify colors; it's simply not the pitch we identify, but rather, the timbre. We can instantly say of a sound, "That's a car horn," or "That's my grandmother Sadie with a cold," or "That's a trumpet." We can identify tonal color, just not pitch. Still, it remains an unsolved problem why some people have AP and others don't.
Levitin also mentions a couple of experimints which might hint that perfect pitch can be learned:
First, they gave non-musically-trained people a tuning fork and gave them a week to memorize its sound. When they came back in for testing, they were "overwhelmingly able to reproduce or recognize 'their' note."
Second, nonmusicians were asked to sing well-known songs like "Happy Birthday" from memory on two different occasions. They "found that although people tended not to sing in the same keys as one another, they did tend to sing a song consistently, in the same key from one occasion to the other. This suggested that they had encoded the pitches of the songs in long-term memory."
For something like language, any adult can learn a second language when they're older, but children seem to learn second languages faster. And it's been observed that it's very difficult or impossible for someone to learn a first language if they're too old. So I guess the question is: is learning perfect pitch like learning that first language, and if you don't learn when you're young, you never can? Or is it like a second language which you can learn when you're older but with more difficulty?