Lennart,
You are correct. Faux is French for false. It can also mean forgery.
There was a rash of Faux Damascus in the late 19th century. The guns were marketed as Damascus barreled. The better makers and the Belgian government stepped in and created laws that prevented any barrels from being marked Damascus or Twist that were actually not. The deeper issue was that these cheap imitations had cold rolled steel barrel with a longitudinal welded seam. When these seams would give, which they did very often, the barrel would split down the length. Not very good for consumer relations. Damascus on the other hand will not fail in this manner. So, it was considered a safer barrel, that people would pay more to have on their guns.
You are located in Sweden? Sweden produced a lot of iron for the gun industry during this period. They had a speciality run for the damascus barrel makers, until the market for damascus declined.
Pete