You use your Strength modifier for the attack roll and the damage roll
The section in the PHB on Damage Rolls states (emphasis mine):
When attacking with a weapon you add your ability modifier - the same modifier used for the attack roll - to the damage.
So the general rule is that the damage roll uses the same ability modifier as the attack roll.
In this case we are given a modified damage modifier (ie it uses Strength instead of the normal Dexterity). In the absence of any other information to the contrary we should apply the above general rule that the damage modifier is the same as the attack modifier.
Thus, in the absence of any other information, by modifying the modifier for the damage roll, the item also modified the ability modifier for the attack roll.
Using this information and looking at the Stat Block of the Creature using the unique longbow provides the answers to your question:
Oversized Longbow. Ranged Weapon Attack: +7 to hit, range 150/600ft., one target. Hit: 11 (2d6 + 4) piercing damage.
The +7 is the +4 strength modifier, combined with the creatures proficiency bonus (+3). the damage modifier is also their Strength modifier (+4).
But there is a second general rule for ranged weapons that we are ignoring here...that ranged weapons use Dexterity for their attack rolls. Why are we disregarding this?
The wording of this item has put us into conflict betwen two general rules, with no clear way to resolve it. The general rules in question are:
Ranged Attack rolls use Dexterity as the to hit bonus (General 1):
You add your Dexterity modifier to your attack roll and your damage roll when attacking with a ranged weapon [...]
Damage Rolls use the same modifier as the attack roll (General 2):
When attacking with a weapon you add your ability modifier - the same modifier used for the attack roll - to the damage.
D&D is an exceptions based game, and the normal way to resolve this would be to use the specific beats general principle. However in this case we do not have enough information to do this effectively (hence the divided debate here!).
As written, the weapon just says damage is 2d6 + Strength. It doesn't say if this is due to the attack modifier having been changed (in the background as a specific exception to General 1) and the damage simply flowing through, or the damage modifier being changed in isolation (as a specific exception to General 1).
Thus we have two options:
Interpretation A: The attack modifier is Dexterity, and the item description is a specific exception to General 2
Interpretation B: The attack uses Strength because the item description is a specific exception to General 1
If we use Interpretation A then we arrive at @AllanMills answer.
If we use Interpretation B then we arrive at my answer.
The consequence of Interpretation A is that this unique longbow is significantly more difficult for the player and the DM to run and not make a mistake.
The consequence of Interpretation B is that the unique longbow is just as easy to run as a regular longbow, but with the ability modifier changed from Dexterity to Strength.
But don't NPCs have different rules to PCs for weapons?
No they do not. In the DMG on page 278 the rules for weapons wielded by monsters are:
If a monster wields a manufactured weapon, it deals damage appropriate to the weapon. For example, a greataxe in the hands of a Medium monster deals 1d12 slashing damage plus the monster's Strength modifier, as is normal for that weapon.
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it's Huge, and quadruple the weapon dice if it's Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12.
A creature has disadvantage on attack rolls with a weapon that is sized for a larger attacker. You can rule that a weapon sized for an attacker two or more sizes larger is too big for the creature to use at all.
From this we see, that if the NPC/Monster is medium size, they would use the normal weapon rules in the PHB. If the NPC is a Large or bigger creature, wielding an appropriately sized weapon for their size (Large creature => Large weapon, Huge creature => Huge weapon) then the damage dice for the oversized weapon are modified.
Should a creature wield a weapon that is too large for it, they either have disadvantage on the attack, or are unable to wield the weapon entirely (barring some feature that would enable them to).