From this well-researched Wiki article:
"Researchers have been trying to teach absolute pitch ability in laboratory settings for more than a century,[60] and various commercial absolute-pitch training courses have been offered to the public since the early 1900s.[61] However, no adult has ever been documented to have acquired absolute listening ability,[62] as all adults who have undergone AP training have failed, when formally tested, to show "an unqualified level of accuracy... comparable to that of AP possessors."
http://en.wikipedia.org/wiki/Absolute_pitch
The interesting thing is that there is a greater prevalence of it in cultures that have pitch-based languages; however, whether that is genetically-related or cultural is unknown.
I have a theory that there is a strong genetic component related to individual neural pathways, and that has to be coupled with learning pitch relationships at a very early age. It's well-known that children who learn an instrument before adolescence tend to be more able to become virtuosi on the instrument, where it is rare that people who learn an instrument later can become great. This is a matter of neural pathways being developed at an early age.
In any event, there seems to be no verifiable relationship between absolute pitch and musical ability.
Incidentally, I do think that lots of people have a sense of tone colors relative to pitch, even non-musicians. For example, it's rare that I'll write an ad track in "C" because over the years I've seen that clients tend to relate to C as a "sad" or "emotional" key, while the relate to D, E and A as "bright" keys, and Bb, Ab, and F as "jazzy" keys. I've often gotten client approval on a project simply by changing its key!
I don't know why this is, but some of it might be genetic, and some cultural.
EDIT: Sorry, Mike, didn't see your post, I was writing this one.