Huh?

you add the modifier to the damage, what made you confused?

The "if you aren't already adding it to the damage" part

is to prevent things like "oh this magic dagger that i have lets me add my modifier into the damage even if i use the light property so i guess i add 3 damage + the 3 of my feat so 6 dmg", they made it as a fail safe for min max probably

Quote from Subtlelogic >>

The "if you aren't already adding it to the damage" part

It means that you don't add the modifier to the damage again if you already have some other feature that lets you do that.

I think anyone with a secondary light weapon can make an extra attack in D&D 2024. Having the Two-Weapon fighting style or feat, I believes then allows you to add ability modifier to damage, and not for anyone else just using a second "light" weapon. I tried to find corroberation of this, but could not find it.

I looked again and found it. It really messes up my dual wielding build, because in order to use two-weapon fighting, both weapons must be light to allow the extra bonus action attack. Frak!

As it says "AN extra attack", does this also work with the additional attack the Dual Wielder feat gives? Since that additional attack is distinct from "the extra attack from the Light property" but still uses the Light property.

My reasoning is that if it were only affecting "the extra attack from the Light property" it would say "THE extra attack".