Does the characteristic for a Lightsaber Technique also determine damage?

Lightsaber damage is not + anything. It's just a straight damage rating, like blaster pistols and grenades. So no, that Characteristic wouldn't normally affect your lightsaber damage.

Lightsaber damage is not + anything. It's just a straight damage rating, like blaster pistols and grenades.

Pretty much this. But all those successes you get on your roll do add to the damage. So it helps.

What about an Ancient Sword?

What about an Ancient Sword?

From page 122, right above Ancient Sword, which is listed as "+2" damage rating: "When wielding a Melee weapon, users add their Brawn characteristic to the damage dealt, unless the weapon’s description indicates otherwise."

So yeah, it adds Brawn to the damage, but uses your Lightsaber skill.

What about an Ancient Sword?

From page 122, right above Ancient Sword, which is listed as "+2" damage rating: "When wielding a Melee weapon, users add their Brawn characteristic to the damage dealt, unless the weapon’s description indicates otherwise."

So yeah, it adds Brawn to the damage, but uses your Lightsaber skill.

And as I said if you are using a lightsaber skill that keys off a different stat the successes on the attack roll add to your base damage of Brawn+2