There is a lot of fuzz over this subject, but here is my 2 cents:
Born with or learned?
What is called perfect (or absolute) pitch refers to the ability of recalling an auditory (mechanical) frequency as perceived by our ears. People trained with our tonal system learn to associate a 440 Hz wave to the note "A", 466.16 Hz as "A#" and so on. Of course this only works in our audible range. Most people have what is called relative pitch, which is the ability of recognizing intervals of two notes. This is more like "frequency subtraction" in the brain, and does not rely on absolute memorization of a frequency, therefore is much easier to learn.
As with many other abilities, some people seem to be better than other, either do to intrinsic factors (genetics, health, etc) or external (musical education, etc). Some abilities are more on the acquired side and some are easy to learn. It is the popular opinion that perfect pitch is an innate ability, and that people "without it" can only perfect their relative pitch, but their brain is just not able to store and memorize an absolute frequency.
Analogy with sight and color
Related to you comment to @ttw, it is funny that you bring the subject of naming colors, as it is somewhat similar. A color is also defined by its wavelenght. What we call "green" is a collection of diverse wavelengths as percieved by our eyes. But what if I say, a specific hue of green that is defined by the exact wavelength of X, would you be able to always recognize this special hue? Our eyes are sensible enough that we can easily differentiate one wavelength from another up to at a certain threshold, defined by the sensitivity of our eyes. Take the same hue and artifically alter the wavelength by a small amount: How big of a change can our eyes percieve? Would you be able to recognize this different color without having the first one as a reference?
In this case, there could be "absolute color observers", people that can memorize the exact wavelength of a color at a smaller range than normal people.
How good are our eyes and our ears depends on physiological reasons. This defines the thresholds of what we can detect as human beings. With that as a basis, we can train and learn for our brain to interpret the signals, but if our detection organs aren't good enough, no learning nor training will be enough.
I hope this gave you an intuition on why some believe that absolute hearing is something you are "born with" while other people believe you can learn it. It is definitely not a trivial answer.
Training to recognize pitches
That said, you can work on your memorization of pitches. Choose songs or pieces that start with a pitch you remember and train yourself to be able to recognize that pitch. For example "La Campanella" by Liszt as a constant D#, or the famous Eb Nocturne by Chopin starts with a Bb. Most people would learn a couple of notes and then find others by the interval (relative pitch). If you want to simulate having a perfect pitch you would need to memorize all 12 tones independently. There is no shortcut and requires a lot of constant practice, but in theory can be done.
You will see there are kids that can do this with ease at very young ages. They are what you would call people with born perfect pitch (obviously learned, but with little to no effort).