How many attacks exactly do I get if I combine the Dual Wielder feat with the Two-Weapon Fighting fighting style?

Question

It says in the Two-Weapon Fighting attack option (PHB, 165):

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.

And then there is the Dual Wielder feat in the PHB:

You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

It somehow gave me an impression that I now may be able to do two attacks in one action, and then the next two as the bonus action, since it says that my weapon productivity literally doubles? Or else, what use can be made out of this addition to the feature?

I also chose the Two-Weapon Fighting fighting style as a Fighter class option, and at that point I am interested only in whether any of these provide me with the right to stab people four times per turn or not.

How many attacks exactly do I get if I combine the Dual Wielder feat with the Two-Weapon Fighting fighting style?

Answer 1

You can take the Attack action once and make another attack as a bonus action.

(Note the uppercase-A vs. lowercase-a attacks, there.)

You can take the Attack action, as one does. Per "Two-Weapon Fighting" (PHB 195) when anyone does that with a light, one-handed weapon they can make a second attack as a bonus action with another light, one-handed weapon in the other hand.

Dual wielder, in the second bullet, basically just erases the word "light" from the two-weapon fighting rule to make it:

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. (Two-Weapon Fighting, modified by Dual Wielder)

So now you can wade into battle with two battleaxes, rather than two handaxes.

Two-Weapon Fighting style adds a modifier to the bonus action attack.

Neither Dual Weilder nor the fighting style change the number of attacks, though: you're still limited to whatever's provided by your Attack action plus one bonus-action attack.

If you want to stab someone four times (and, really, who hasn't been there?) look additionally toward Extra Attack, Action Surge, haste... those cover 99% of the ways you could get to a 4th attack.

Answer 2

Normal Attack action plus one as a bonus action

The Dual Wielder feat does not grant any additional attacks. Its main benefit is allowing you to use Two-Weapon Fighting with non-light weapon, thereby increasing your damage output. It has the additional benefit of allowing you to draw two weapons at a time, allowing you to get around the one free item interaction per turn limitation of drawing multiple weapons.

The Two-Weapon Fighting Style also does not grant any additional attacks, simply allowing you to add your ability modifier to the damage of your bonus action attack.

With this feat and fighting style you can make one Attack action, and one additional attack with a bonus action as granted by the two-weapon fighting rules. When you gain the Extra Attack feature you can make more than one attack with your main action. However you can still only make one additional attack as a bonus action regardless of how many attacks you can make with your main action.

It somehow gave me an impression that I now may be able to do two attacks in one action, and then the next two as the bonus action, since it says that my weapon productivity literally doubles?

This part of your question seems to be misreading the rules. Nowhere does it say that your "weapon productivity literally doubles", it says you can draw two weapons instead of one. Not that you can make additional attacks with them.