How does weapon durability work in Fates?

Correct, weapon durability does not exist in Fates (for any path - Birthright, Conquest or Revelations). Staves however, do have limited number of uses; the stave will go away after it's all used up. In the Birthright and Conquest paths, there is no way to combine Staves into a single item with more uses; staves can be combined in Revelations.