The Dueling Fighting style description says:

When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

The Bracer of Flying Daggers magic item (from Waterdeep: Dragon Heist, p. 190):

This armband appears to have thin daggers strapped to it. As an action, you can pull up to two magic daggers from the bracer and immediately hurl them, making a ranged attack with each dagger. A dagger vanishes if you don't hurl it right away, and the daggers disappear right after they hit or miss. The bracer never runs out of daggers.

If you have the Dueling fighting style, your hands are empty, and you activate the Bracer, do you get the damage bonus to both throws?

The question is whether thrown melee weapons qualify for Dueling or not

The description specifies that you pull daggers out of the armband (despite the logistically problems of sticking a full-sized dagger in an armband. I would have gone with knives had I designed it), which are on the melee weapon table and therefore are melee weapons (magical ones, no less). You obviously don't have another weapon in your other hand, so all we need to know is whether throwing a melee weapons counts as "wielding" the weapon.

There is no game-definition for "wielding", so we rely on common English to understand it. Google gives this definition for "wield":

hold and use (a weapon or tool).

Under that definition, and in my own opinion, I'd say that throwing a melee weapon counts as wielding it for the purposes of Dueling.

Crawford agrees that Dueling applies to thrown weapons, for what that is worth (keeping in mind his Sage Advice tweets are unofficial).

The related question Does Dueling Fighting Style apply to thrown weapons? has arguments for and against this ruling.

This is a poorly worded item

It is unclear whether you are drawing both daggers at once in both hands, or one at a time, or what. For some interpretations it would matter whether you are drawing both at once, or if you are throwing both at once, but this is mine: I assume you are drawing one, throwing one, then repeating. If you draw one, immediately hurl it, then draw another and immediately hurl it, you've still drawn 2 daggers and immediately hurled them.

Only to one throw, not to both

Short of contradicting evidence, the Daggers summoned by the Bracers of Flying Daggers are in every respect identical to regular, PHB Daggers, except for the added properties that they vanish if not thrown or after they are thrown, and you are allowed to make two attacks with them regardless of your [in-]ability to use the Extra Attacks feature.

Because of this, a character would obviously benefit from their Dueling feature: Daggers are a one-handed weapon, can be thrown with a single hand, and thrown weapons benefit from the Dueling feature*, so you should get the Dueling bonus.

There is a catch though:

Dueling. When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

If you draw two daggers from the bracer, you'd be holding two weapons, which violates the requirements of the feature. So at the very least, the first dagger thrown would not benefit.

The feature seems to imply that both are thrown at the same time, meaning neither dagger would benefit, but if your DM decides there's a "happens-before" relationship between the two attack rolls (i.e. one definitely happens before the other), then the second dagger would probably benefit.

This becomes moot, however, if you draw only a single dagger and throw it, since the requirements of Dueling would then be satisfied—and since the bracer says you may draw "up to two" daggers, you're fully allowed to draw only a single dagger.

^{* the top answer in that post contradicts RAI, and I recommend looking at the second answer instead.}

Dueling will benefit in this situation depending on the interpretation you use

The Dueling fighting style states:

When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

Daggers appear on the melee weapon table and the dueling fighting style requires a melee weapon. It does not say that it requires a melee weapon attack. This sounds like we're on the right track but, in throwing the melee weapon to make a ranged attack, it is no longer in your hand.

One argument is that the dagger is in hand at the moment the attack is initiated. However, the attack does not become a ranged attack until the instant the weapon leaves the hand. Therefore, the dagger not being in the hand is integral to it being a ranged attack (not a melee attack) and, as such, Dueling would not benefit the knives if they are thrown.

That all said, Jeremy Crawford has stated in a (now unofficial) Tweet that you can interpret the Dueling to allow the bonus to apply to ranged attacks. This is outside of the PHB but it's from the lead rules designer, so one could make a case either way and have strong support for the argument. As such, it probably depends on which argument the DM finds more persuasive.