I've seen the photos, so I know he told me the truth. The only extra information that I can add is that he came from a well-off family and was attending Princeton. My guess is that those factors had something to do with it, but that is just a guess on my part.
Most likely he was a graduate of what was known as the Naval Aviation Cadet Program.
It was designed to pump out Naval Aviators quickly as the need grew starting in the mid 1930's, and continuing all through WW2.
The requirements were only that the applicant be between 19 and 25, have an Associate's Degree or completed 2 years of a Bachelor's. Applicants could be Civilians or Enlisted Sailors and went through the normal flight training pipeline (about 18 months) and then had to spend usually about 3 years on Active Duty during which time they could eventually be commissioned straight to LTJG in the Navy Reserve.
The Ivy League was actually a hot bed of the early Navy college programs, as the NACP operated essentially in parallel with ROTC.
Princeton University was home to a unit of the V-12 Naval College Training Program. Upon completion of this program, an officer commission in the Naval Reserve was offered. It is possible that is how your father gained his officership in the active-duty United States Navy at the age of 20. From there, he would have completed his training as a fighter pilot.
Source: Richard D. Challener, "The Response to War," Princeton History (1992) pp 48-65
With a little googling, it looks like he actually flew a Helldiver dive bomber and he was shot down--he survived. If you have anything to share about these planes and the role they played in the Pacific, I would love to hear it.
Since your question is not limited to just the Allied pilots, here is a perspective from the Soviet side, collabed from pilot accounts, stories by veterans on iremember.ru and Russian documentaries:
At the beginning of the war, the USSR had a large amount of properly trained pilots from aviation schools all around the country. Large numbers of these trained combat pilots were lost in the first months of the war, leaving a (small) surplus of material and a shortage of pilots.
Most Russian pilots were young students from the cities and komsomolists of the era. The requirements, much like /u/DBHT14 noted for allied pilots, were based on age, eyesight and intelligence. Age differed a bit from the Allied requirements, as it was lowered after the massive losses of qualified pilots in the first months. People as young as 16 are said to have been admitted to pilot training. There was no upper age limit as far as I have read.
During the first years of the war, with extremely high demand for pilots, the educational program underwent a few cuts and changes to allow for faster training and more flight hours. Where in 1937 pilots would have spent about 2-3 hours in the air on a daily basis, during the war some would spend more than 6 hours flying, every day. This led to some pilots being thrust into combat situations after as short a training period as a few weeks(the average training period during peace lasted more than two months as a soviet ace mentions).
The combination of lowered standards to replenish the losses suffered at the start of the war, which were staggering(exact numbers are incredibly hard to come by as some are simply reported missing, but some reports claim over 40% of combat pilots lost within the first 6 months of the war), meant that both the age limits and the training duration went down. That's how a 20 year old, and far younger boys and girls got to be pilots in WW2 on the Soviet side.