They are in fact teenagers by the modern definitions. You were a full responsible adult shortly after puberty. Many of the greats did their best work before 20.
Who did their best work before 20? I looked up a bunch of Greek philosophers, and it doesn't seem to apply to any of them. Socrates was in the military during his early life, and Aristotle did his most famous work late in life.
One possibility is that GP is thinking that "life expectancy was 25, so 20 means you're an old geezer on the deathbed." Alternatively, it may also be a reference to the extreme youth of Alexander the Great, who became king of Macedon at age 20 and proceeded to put down a Greek rebellion and then conquer Thrace, Persia, Egypt, Bactria, cross the Hindu Kush, conquer bits of India (then unknown to the Greeks) and was only stopped from going further when his army said "it's been long enough, can we go home?" He died 3 years later, at age 32. Obviously, he is very much an outlier.
Measures of life expectency are misleading in this regard.
Much of low life expectency comes from very high infant mortality -- deaths to age 5 or 10. Once you've passed that threshold, your odds of survival increase markedly. Perhaps slightly lower than today's first-world standards, but hale-and-hearty at 15 doesn't mean you're going to fall over dead at 25.
You also want to look at life expectency at any given age. And I distantly recall that the number of (I can't remember specifically which) Greeks or Romans surviving to age 80+ compared favourably to modern times.
I remember reading stories where he charged ahead of the rest of the army, scaled enemy walls, chased down royal bodyguards and so on. He was bloody crazy. A guy with his (lack of) brains should not have made it past 16, let alone get to 32.