Anytime something says it grants Melee +X, you should assume it means the Melee +X trait and that it works exactly like the description of that trait in the Codex. Meaning, it only applies on the first possible attack of an attack action. You can contrast this with something like Johktari Hunting Knife which says: "If this Mage is level 5 or greater, Johktari Hunting Knife rolls an additional attack die." This wording indicates that the base number of attack dice has actually been increased rather than just receiving a bonus.