How does Finn Edwards' passive ability affect the directive to discard Unique Assets?
According to the Reference Manual for Eldritch horror, when Discarding: 


  When a double-sided card is discarded, it is immediately shuffled 
  back into its respective deck.


Nothing supersedes this rule in updated manuals or contradicts this anywhere else within the guide.  For instance, under the Discarding heading in the Reference guide:


  Whenever a card is discarded, it is placed in a faceup discard pile 
  by its deck.
  
  Double-sided cards, such as Spells or Conditions, are immediately 
  shuffled back into their respective decks when discarded.


This still indicates that while the general rule is a separate discard pile, double-sided cards as a whole are immediately reshuffled.

For how this works with Finn's passive ability, we need to look at the specific wording of his character sheet


  Effects cannot cause [Finn] to discard ...


Emph added.  So this means that a game effect (e.g. an encounter causing Finn to become delayed and lose an ally) can be ignored.  The natural consequences of using an item, however, cannot be ignored.

Thus, Finn's ability would prevent someone from stealing his Old Journal, but if said journal were aged and worn, he would be unable to prevent its falling apart upon reading it.