The reason it says your attack deals weapon extra damage is to try to drive home that it is the same damage as the weapon. Just like adding your strength to the damage of an attack is

P.196 PHB Damage Rolls section states. "Each weapon  ,spell, and harmful ability specifies the damage it deals. You roll the damage dice add, Any modifiers, and apply the damage to your target.

Above shows that your weapon deals a specefic damage. You add your fighter level to this damage.

Note that this is only important for resistances, vulnerabilities, and immunities.

Xanathars attempted to be clearer on effects like this and on the Gloom Stalker rangers additional dice to weapon damage to attempt to avoid the scenarios of confusion that sneak attack and Collosus Slayer have.

The reason it says your attack deals weapon extra damage is to try to drive home that it is the same damage as the weapon. Just like adding your strength to the damage of an attack is.

On PHB p. 196, the Damage Rolls section states: "Each weapon, spell, and harmful ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target."

The above shows that your weapon deals a specific damage. You add your fighter level to this damage.

Note that this is only important for resistances, vulnerabilities, and immunities.

Xanathar's attempted to be clearer on effects like this and on the Gloom Stalker rangers additional dice to weapon damage to attempt to avoid the scenarios of confusion that the rogue's Sneak Attack and the Hunter ranger's Colossus Slayer have.