A mop can safely be used on hardwood floors but only if the mop is just damp and not saturated.
So if you use a mop to clean your hardwood floors then make sure you wring the mop out to where it's just a little wet and not sopping wet.
If the mop is too wet then you'll create a puddle of water that will be absorbed by the hardwood floors which can then crack and warp the hardwood floors.
So as long as you don't leave any standing water on the floor then mopping the hardwood floors will be just fine.