Since having 2 turns in a 2-3 player, does the ability of a unique district trigger for each character turn?
For example, I have the Laboratory in play, can I trigger the ability to discard 1 card to gain 2 gold, for each character I play?
Yes you can use the unique ability twice.
The rules here state the following regarding 2-3 player games.
each player has two turns each round (one turn for each character).
So if a card says it can be used 'once per turn' then you will get to activate once during the turn of each character you took.
The rules make a distinction between 'round' (The process from character selection and calling each character in order) and 'turn' (when a character takes and action and/or builds a district and uses special abilities.