The character does use both weapons, yes. He uses one to attack. He can use the other one for a second attack (normal two-weapon fighting) or to give himself a defensive advantage (Two Weapon Style maneuver). In the latter case, I don't take it to mean that he is strictly parrying, but just using the weapon to improve his defensive posture in general. Sometimes that might simply mean that he has an easier time fending off his target, so that he can focus more on his overall defense. More generally: Without the second weapon you don't dominate your target enough to fully defend against everyone else.
Nothing in this rule talks about dominating your target and overwhelming it. By the rule, you only dominate your target if you successfully beat it on the attack test, or force it to spend all its actions into full dodges so it can't take an action to place an attack, otherwise you don't dominate it at all.
But, according to the rule interpreted as you do, in both case whether you dominate it or not, you don't have to spend a complex action to use the full defense with your other weapon, so the bonus you get is obviously not about dominating your opponent or not.
Besides, would it be just about dominating your opponent and not about using actively your second weapon for defense, then a character with only one weapon could use it : he also could dominate his opponent with one weapon, exactly as the TWS user do, and would then be able to concentrate more on his defense, again exactly as the TWS user do.
And if overwhelming a melee opponent grants you free full defense against all others opponents, including those who shot you from distance, who are obviously not overwhelmed by your attack since you don't attack them, then this would allow you to have a free full defense against all opponents even if you don't attack anybody in melee.
Then, since making an attack with a gun is a lot more simple and takes you less time, both in the shadowrun RPG conventions and in the shadowrun ruleset, a character who's able to concentrate on its defense while attacking with a melee weapon should obviously be able to concentrate on its defense even while using a gun.
It's straight logic : your interpretation leads straight to extending this maneuver's effects to ranged fight.
And can be extended to all kind of actions which aren't more complex than a melee attack.
I don't remember if there were any major changes between the SR2 and SR3 melee ruleset, but in SR3, melee was just overkill. You just had to take a weapon with the best reach you could, a good strength, and you blasted all. A troll with a +2 reach weapon, it was clear overkill.
I find melee in SR4 still very powerfull, but much more balanced.