What happens when a player controls multiple identical monsters?
Since you didn't specify which specific game in the Dungeons and Dragons Adventure System you're referring to, the following excerpt is from the Wrath of Ashardalon rulebook


  Place a Monster on the new tile. To place a Monster, draw a Monster Card and place the corresponding Monster figure on the scorch mark of the newly placed Dungeon Tile. (The name of each Monster is printed on each figure’s base.) If you draw a Monster Card that matches a Monster Card already in play in front of you, discard that Monster Card and draw again. If another player has a matching Monster Card, however, that’s okay (emph added)


This states that you cannot have this situation, as you should have discarded the matching card in favor of a re-drawn, unique (to you) monster.

What this doesn't answer, however, is when there is an option to pass a monster or being forced to pass a monster.  Looking through the rules of both old (i.e. Wrath of Ashardalon) and new (i.e. Temple of Elemental Evil), I don't see this directly addressed anywhere, even in FAQs.  However, I think that we can refer to the spirit of the rules rather than the letter of them in order to resolve this.

One concept that does occur frequently within the rules/FAQ is that no single player may have more than one copy of a monster.  So, if you attempt to pass a monster to the right, but your right neighbor has a copy, either that pass is forfeited or the attempted pass continues further to the right.  Given that there are monster cards that force passing, I believe that the intent would be to continue the pass further around the table.  If everyone else has a copy, then ultimately it winds up being passed back to the original owner.