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.