I don't think Ardyn's actual sword in the main game is sword of the wise. The description especially doesn't add up.
I'm under the impression the building of the wall was something that happened leading up towards Regis and Noctis's time, when civilisation was becoming more advanced, not back in the old days when Ardyn was still mortal and the world still looked medieval somewhat (as hinted by the Ardyn concept art of him riding the chocobo)
He's got all the Armiger weapons, save for the sword of the father, so perhaps they just gave him that sword because, well he needs to have a weapon in that game and they just went with that
We don't know what his default sword is.