The Dragon class seems to use the same attack for the regular attack, which calculates damage based on Strength, and the special attack, which calculates damage based on Intelligence. Is the Dragon's attack damage based on Strength, Intelligence, or some combination?
I dug through the code to find an answer to this, and it turns out both buttons for the dragon class do the exact same thing, which is to call
CastSpell() (normally only done when pressing the 'spell' button).
CastSpell(), the total damage is only affected by
PlayerObj.TotalMagicDamage - which depends only on your Magic-Up ("Intelligence") skill and any Magic-Up fairy chests you've opened - and some class/spell-related constants.
Thus, increasing your Attack-Up ("Strength") will not increase the dragon's attack damage at all; only Magic-Up will.