Harrison Ford got a scar on his chin from a car accident he had in real life, but is there a Star Wars in-universe explanation about Han Solo getting it?
I believe it is explained in the book Han Solo and the Lost Legacy by Brian Daley. We see him getting cut across his chin during a fight.
The chief took a moment to slash at the pilot. Han suddenly realized how impetuous he had been; as if another man entirely had occupied his body for a moment. He didn’t quite avoid the stroke and the knifepoint cut across his chin.
[ later ]
“Don’t fold until the hand’s over, ” he encouraged her quietly. She turned to him with a labored smile, brushing her dirty fingers against his stubble of beard, tracing the raw scar across his chin. “You know, this is an improvement, Solo. You’re not Slick now, not so smooth and careless.”