Yeah, sort of.. Does Lindenroot then have to discard 2 cards (or exert a hobbit) if the shadow player assigns a minion to him?
Or does that only happen if the free peoples player assigns him?
That only happens if you use that assignment action. If you don't and
Lindenroot happens to be assigned some other way (for example,
Between Nazgul and Prey, a
Counts But One variant that would work for Ents, or in your example) then nothing happens.
It would be as you suggest if it said "Each time this companion is assigned to a skirmish, discard 2 cards from hand or exert an unbound Hobbit". But as it's an assignment action, you can either use it or not.