Reading from PHB 5e, I like to know if I choose a Fighter class and pick Two-Weapon Fighting as my fighting style and later in the game if I choose the feat Dual-Wielder. Is picking Two-Weapon Fighting obsolete if Dual-Wielder is superior effects than two-weapon fighting and is two-weapon fighting a requirement for Dual-Wielder?
If Dual-wielder had a prerequisite, it would say so right under the name of the Feat. Since there's nothing there, that means you can pick it with or without the Two-Weapon Fighting style.
But they do different things and will stack with each other; picking both is thus extra powerful. The Feat will allow you to use larger weapons (as well as give a bonus to AC) while the style lets you apply your ability modifier to off-hand damage. Having both means you get to use larger weapons and use your ability modifier to off-hand damage.
Since it adds damage modifiers, Two Weapon Fighting is not Obsolete.
Two Weapon Fighting allows you to add your damage modifier to your off hand/other/second weapon, though the fighting style is limited to light/finesse weapons. You are not required to have this fighting style as a pre-requisite for Dual Wielder (there are no prerequisites for that feat) but it is beneficial to use that fighting style.
Two-Weapon Fighting (Fighting Style, PHB, p. 72)
When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.
Two-Weapon Fighting (That anyone can use)(PHB, p. 195)
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
Dual Wielder (feat, PHB, p. 165) offers different features:
- You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.
- You can use two-weapon fighting even when the one handed melee weapons you are wielding aren’t light.
- You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.
From bottom to top, there are three cases of attacking with two weapons:
- You can use two light weapons, but the second one gets no damage bonus (anyone)
- You can use two light weapons, and the second one gets the damage bonus (Two Weapon Fighting, the fighting style)
- You can use any two weapons, get a +1 AC bonus, and can draw or stow both with the same action/interaction. Unless you also had the Two Weapon Fighting fighting style, this feat won't add the damage modifier bonus to the second weapon.
The Two Weapon Fighting style makes your Dual Wielder a bit more effective since it adds the damage modifier to both attacks, not just the first one.