You can wield & attack with two weapons without any special class feature or feat.
Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):
Two-Weapon Fighting
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.
As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.
Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.
The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.
Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.
According to the RAW in order to use a shield (6 lbs) and a dagger (light) weapon to attack in the same round, you would need the feat Dual Wielder.
The rules for two-weapon fighting (not the fighting style) on PHB page 195 stipulate that to get a second attack both weapons used must be light. In the weapon descriptions, the heaviest weapon that is classified as a light weapon weighs 3 lbs.
The Two-Weapon Fighting fighting style just allows you to add your stat bonus to your ability modifier to the damage from your second attack.
The Dual Wielder feat allows you to use weapons that are not light.
For clarification purposes, you must have the Dual Wielder feat to get both attacks when using the two-weapon fighting style if both weapons are not light melee weapons. (That is to say, if one or both of the weapons is not a light melee weapon then you absolutely need the Dual Wielder Feat to get the bonus attack.)
Further clarification, according to the section on two-weapon fighting (not the fighting style) in the PHB page 195 (emphasis mine):
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 you are holding in the other hand.
In other words, to get the off-hand bonus attack both the weapons you are holding must be light melee weapons if you are only using the two-weapon fighting style without the Dual Wielder feat. The feat allows you to get the bonus attack even if the weapons are not light melee weapons. So in the above question, a dagger is a light melee weapon but the improvised shield is not, therefore the Duel Wielder feat is needed to attack with both weapons in the same round.
The addition of the Two-Weapon Fighting fighting style would allow you to add you stat bonus to the damage of the second attack.
Best Answer
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.