I do not possess this gene, but in my musical life I've known a handful of people with perfect pitch. More than one of them mentioned that it is both a blessing and a curse. One of them, a friend who is a gifted bass player, said that it can be maddening at times, living in a world that wheezes, whistles, bellows, and mutters as it does. Because most sounds are no where near perfectly pitched, he would often find himself obsessively noting that this sound or that sound was almost right, but off by just so much, clashing abrasively with some other sound. The upside, though, was that he never, ever played out of tune, he could readily play numerous instruments surprisingly well without practice, and he could easily recall and analyze in detail complex melodies and chord progressions. He is a successful commercial real estate agent.
There are, in fact, teaching methods for sale that claim to be able to teach people to have perfect pitch. While that may well be possible, it is far more common among musicians to develop a heightened sense of relative pitch which, I've always believed is based on memory. My wife and son both have excellent pitch memory. When they sing a song, spontaneous and unaccompanied, they almost always nail it in the original key. I am not as good a singer as they are, but I am usually in the ballpark. These are handy skills and, while not intrinsically difficult to attain, typically come only with practice and experience.
A related skill that I have is that I can tune my guitar, both electric and acoustic, with out a pitch standard, even without being able to really hear the instrument. It is not "perfect", but it is usually close. I can sense by the way the strings feel, especially the A string. And once the A feels right, it is pretty easy to get the rest close. What my brain is doing in this case is responding to the tension of the string and the way it feels under my fingers when it vibrates, so it is more like muscle memory and less like an aural experience. I don't think this is an unusual skill, per se, and there are probably many guitarists who can do this. I developed it in my 20's when I worked in a couple of different music stores in Texas for several years. I have restrung and tuned many more guitars than most people, and I'm sure that is the source of this peculiar ability. It is all about familiarity and conditioning and I'm sure many other people in other disciplines have similar experiences.
The way we process and experience musical information and respond to sound is a never ending source of wonder and fascination, though. And it never ends. It is science, it is emotion, it is spiritual, and it is experiential. We will never figure it all out or reach the end of what there is to hear. There is infinite variety in musical expression...it is a glimpse into something beautiful, something personal, and something eternal. I understand only a few very small pieces of it, but I can't put it down. And I don't want to.