What's the name of this co-op card-based game?
Sounds like The Grizzled or in its native language, Les Poilus.

An excerpt from a description I found


  Players take on the role of French soldiers in World War I. You can
  choose your difficulty level, which means how many cards are in your
  hand. On your turn, you can either play a card from your hand or
  withdraw. There are two types of cards; ones with symbols on them
  (night, rain, bullets, gas mask, etc) and ones with a lightning bolt
  and text on them. If you play a symbol card, it goes into the river.
  If you ever play a card that would cause you to have three of the same
  symbol, you lose.