[RPG] Can the Soulknife’s psychic blades be used with extra attack as a thrown weapon

At least by the rules as they are written the dueling bonus applies to both attacks. On page 24 of the SRD the Dueling Fighting Style text is:

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.

Note that the bonus to damage is activated immediately. So, if you lose one of your psychic blades and immediately gain another you get that bonus applied.

Not adding the modifier is a special feature of Two-Weapon Fighting.

The rules for damage say:

When attacking with a weapon, you add your ability modifier — the same modifier used for the attack roll — to the damage.

Notice there is no mention of this being an attack action or attack with a bonus action. This is the general rule for making an attack.

In contrast, Two-Weapon Fighting has a specific rule:

You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

This rule is specific to Two-Weapon Fighting - it creates an exception to the general rule for damage rolls.¹

So you add your modifier to the damage of Soul knife’s bonus action attack since nothing in the ability contradicts the general rule of adding your ability modifier to the damage.

^{1 The game has a "specific beats general" rule in the introduction to the Player's Handbook which states: "That said, many racial traits, class features, spells, magic items, monster abilities, and other game elements break the general rules in some way, creating an exception to how the rest of the game works. Remember this: If a specific rule contradicts a general rule, the specific rule wins."}