Is Jenga a cooperative or competitive game?
Jenga, as designed, is most certainly a competitive game. The rules released with the game have stayed consistent over the years and have never described how to play in a cooperative manner.

Most versions of the rules (such as the 2000 version) have a "Solo Play" variant that might also be able to be played cooperatively with multiple players:


  Play alone for practice! Can you top your record height before the
  tower falls?
  
  
  Carefully remove one block from any level. Use a steady hand to stack it on top.
  Continue to remove and stack blocks to build the tower taller…and shakier.
  How tall is too tall? Find out when the tower falls!
  


One Jenga review described playing a variation of this during a team building event where teams of 3 played cooperatively for a high score:


  I first encountered Jenga in a corporate team building activity. The group was split into teams of three. Each team was given a Jenga set. The objective of the game was for each team, following the standard Jenga rules, to try to build the tower as high as they could. The team that built the tallest tower, without the tower collapsing, won the game.