The funny thing is, Greenbull's sword being made out of wood isn't even a legitimate argument to counter it being a sword. Weapons are named for their function, not their material.
Take a mace. A mace is a melee weapon with a weighted head designed primarily to inflict blunt force trauma.
A mace can be made entirely out of wood, entirely out of metal, have a wooden handle with a metal head or a wooden handle with a stone head. As long as the function is the same and it can perform said function then it doesn't matter what it is made out of, it is still a mace.
Same with bows. They can be made of wood or carbon fibre. Cultures with limited access to wood had to get creative with materials like animal horn and sinew. Regardless of what they are made out of, they are ranged weapons designed to fire arrows. Again, bows are named for their function, not for what they are made out of.
Meanwhile in One Piece, you have a man that can make biscuit as hard as steel and then precisely recreate his metal sword using biscuit equal in strength to steel. He then uses that weapon exactly like his metal sword because it is functionally the same.
That isn't a "fake" sword, that is just straight up a sword.
If biscuits can be sharp enough and strong enough to be used as swords in this universe, then so can wood. It would still be a sword.
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