When a dragon PC ages does it work like getting a paragon class that stacks with sorcerer levels? For example when a brown dragon advances from wyrmling to very young, its strength increases by 4 and its constitution increases by 2. Would a player character get +4 strength and +2 constitution upon aging?
Draconomicon includes information for dragon player-characters—like aging effects—on pages 141–4. Such rules are too extensive to reproduce here, but, in short, "When a dragon reaches a new age category, its ability scores improve across the board," but, as "it ages… the dragon is required to devote a level every few years to its dragon 'class,' reflecting the extra Hit Die or level adjustment it gains from aging." Draconomicon mentions on page 144 how to determine what the game thinks is an appropriate level adjustment for an aging brown dragon (Monstrous Compendium: Monsters of Faerûn 38–40).
So, yes, as dragons age, their ability scores increase, but the game believes such ability score increases shouldn't come free.