I'm somewhat acquainted with the dueling that was popular in German universities from the Romantic era on through the Nazi era, but this description, which I found online says it best:
"Mensur" refers to a custom that began in German, Austrian, and Swiss elite universities in the mid-1800s and lasted until the mid-1900s. It had to do with 'beautiful scars' (Schmisse) resulting from highly-regulated fencing duels. These cheek scars were seen as a rite of passage, a symbol of manhood

You can see he has a scar in other photographs which have nothing to do with
Ship of Fools
, though. At least that's what I noticed in a quick Google Image search.
.

The character in Ship of Fools, Dr. Schumann, got his scar when he "tried to be a man", ostensibly a duel or something.
The actor, Oskar Werner, who played Dr. Schumann, go his scar in the makeup department.