If you use True Polymorph to change into a young or adult dragon, and sustain the spell for the full hour so that it becomes permanent, will you (over the following centuries) mature into an ancient dragon? Or will you stay as the form you polymorphed into?
True polymorph will "turn a creature into another kind of creature." Before we get into the weeds, let me emphasize this: the target is now a different kind of creature.
Later in the creature into creature section we see some explicit calling-out of what is changed by the spell: specifically the creature's statistics "including mental ability scores, are replaced by the statistics of the new form. It retains its alignment and personality." (PHB p.283, emphasis mine.)
So, don't only the statistics change? And "aging" isn't a trait in the stat block....
A creature's "statistics" are spelled out exactly on MM pp.6-10. So we know explicitly that size, type, AC, saving throws, reactions, &c. are all assumed. You seem to be worried that "the natural propensity for aging" isn't on that list. And because it's not enumerated, it doesn't change. But I remind you: the target has been changed into that kind of creature.
This language--the "kind" of creature--is necessarily more-expansive than just what's listed in the stat block. This means the target has become a young dragon along with all that entails; not just what the MM entry says. MM doesn't say that dragons breathe or have circulatory systems, but we know they do and the polymorph target will, too. MM doesn't say that a dragon ages, either; we don't need that "trait" of a dragon enumerated in the stat block, we know it to be so.
Now, in case this all seems a bit wishy-washy, please refer back to the emphasized portion of the spell text above: retaining "alignment and personality." Alignment is a statistic. Personality isn't. Thus the spell text itself tells us that we need to think about the totality of the being--statistics and otherwise--when understanding the transformation. We wouldn't need to call out personality not changing if there weren't a possibility that it would change, even though it's not a statistic.
"Personality" is a tricky term in 5e: it gets used in some places where we might wish it were defined, yet it is not. See many of these questions on the matter.