You can have what you want when you get to 4th level
You want a Dragonborn who fights with two longswords. OK, here's how to make that work. If you start as a Fighter at level 1, you can choose Two-Weapon Fighting for the Fighting Style feature (PHB, p. 72):
When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.
But that only works with light weapons, per the general Two-Weapon Fighting rules. You can do the same thing as a Barbarian, but your second weapon won't get the Strength bonus to damage, since you won't get the Fighting Style feature.
The general Two-Weapon Fighting rules, which any character can use to wield two weapons at the same time, state (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.
If either weapon has the thrown property, you can throw the weapon,
instead of making a melee attack with it.
At level 4, you gain the Ability Score Improvement feature, and can take the Dual Wielder feat (PHB, p. 165):
You master fighting with two weapons, gaining the following benefits:
- 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.
Once you gain this feat, you can upgrade the two light weapons you've been wielding to longswords.
You will be using your bonus action to attack with the second weapon in any case.
To get this at level 1, switch race to Variant human
If you are willing to switch your PC's race to Variant Human (PHB, p. 31), you gain a feat at character creation. This means you can wield two longswords (by choosing the Dual Wielder feat) and the same Two-Weapon Fighting fighting style, starting at level 1.