The Vampiric Bite, from the new Dhampir lineage race, says as follow (emphasis mine):
"Your fanged bite is a natural weapon, which counts as a simple melee weapon with which you are proficient. You add your Constitution modifier to the attack and damage rolls when you attack with your bite. Your bite deals 1d4 piercing damage on a hit. While you are missing half or more of your hit points, you have advantage on attack rolls you make with this bite."
Normally, a simple melee weapon attack works as follow:
To make an Attack roll, roll a d20 and add the appropriate modifiers.
Modifiers to the Roll
Ability Modifier The ability modifier used for a melee weapon Attack is Strength, and the ability modifier used for a ranged weapon Attack is Dexterity.[...]
Proficiency Bonus You add your Proficiency Bonus to your Attack roll when you Attack using a weapon with which you have proficiency, as well as when you Attack with a spell.
This translates into rolling 1d20 + STR modifier + Proficiency bonus.
Does it mean that a Vampiric Bite attack adds CON modifier to the above formula (1d20 + STR mod + Proficiency bonus + CON mod)?
Same question for the damage roll:
When attacking with a weapon, you add your ability modifier—the same modifier used for the Attack roll—to the damage.
Normally for a melee weapon the attack formula is [weapon-damage-die] + STR modifier. If another modifier is used, instead of the usual one, the wording is different from what we see in the Vampiric Bite. An example is Finesse weapons, which say as follow:
When Making an Attack with a finesse weapon, you use your choice of your Strength or Dexterity modifier for the Attack and Damage Rolls. You must use the same modifier for both rolls.
Is it correct to say that, unlike Finesse weapons, the Vampiric bite does not replace the STR modifier with CON, but rather adds both (making the damage formula 1d4 [weapon-dmg-die] + STR modifier + CON modifier)?