While the ‘Damaged’ tattoo went unexplained in Suicide Squad, David Ayer revealed after the movie’s release that Joker got it after his teeth were smashed out by Batman in retaliation for Joker killing Jason Todd, a.k.a.
Ayer also noted ahead of Suicide Squad’s release that he gave Joker tattoos is because he took inspiration from “drug lords on Instagram.”
That, combined with the more modern comics influence that Ayer mentioned in his Twitter post, and you get the DC Extended Universe’s Joker, who’s been clashing with Batman for years by the time Suicide Squad kicks off.