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.
You’re missing the two-weapon fighting combat option rules. Without those, the feat et al. won’t make very much sense.
The two-weapon fighting combat option allows you to get an extra attack when you full attack, but all of your attacks in that full attack take a penalty. The extra attack has to be with a different weapon from your regular attack (“offhand” weapon), and the penalties you take depend on whether or not that offhand weapon is light or not.
The Two-Weapon Fighting feat reduces those penalties to a point that this option is actually usable.
Another key thing to realize is that having two weapons in hand does not automatically mean you are using the two-weapon fighting option. You have to choose to use it, to get an extra attack and take an attack penalty.
As for the +6/+1 thing, that’s called an iterative attack, and that’s free just for having BAB that high. It is separate from, and can be combined with, the two-weapon fighting option. So, assuming you have the Two-Weapon Fighting feat and a light weapon in the offhand (the only case in which the two-weapon fighting option is a good idea), with +6/+1 BAB you could attack normally, for +6/+1, or you could choose to use two-weapon fighting, taking a −2 penalty. Then your attacks would be +4/+4/−1, where the second +4 attack has to be with the offhand weapon. If you take the Improved Two-Weapon Fighting feat, that becomes +4/+4/−1/−1, because Improved Two-Weapon Fighting gives you a second bonus attack with the offhand weapon, at −5 (matching the penalty on the iterative attack).
Best Answer
In D&D 5E, specific rules take precedence over general rules. Since the two-weapon style, available through the fighter and ranger classes, is a specific rule, let's start by reviewing the general rule:
According to this rule, attacking with a second weapon requires :
Note that these rules do not specify that you must attack the same target, nor that the weapons used need to be identical. Normally, you would add your ability modifier to the damage of your attacks. However, these rules also include a specific ruling that only applies to two-weapon fighting :
You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
This does not normally affect your attacks granted by the attack action, only those granted by this specific use of a bonus action. This, however, is then affected by the specific rules introduced by the two-weapon fighting style:
As such, a fighter with this fighting style can apply his ability modifier on all of his attacks, including off-hand ones. However, he is still subject to the general ruling stating that he must wield two light weapons to do so.
To wield weapons that are not light, one would require the "Dual Wielder" feat, found in the optional feat rules presented in chapter 6 of the PHB. This would allow a fighter or a ranger, for instance, to wield a pair of longswords or a longsword and a battle axe. This feat also has the added perk of allowing the drawing and sheathing of both weapons as a single item interaction, as well as granting a +1 to AC while wielding two weapons.
TL;DR