So through these stories, the Joker is saying his scars weren’t just physical, he’s connecting them to an emotional trauma. He does the same thing to Harvey Dent -- implementing physical scars that are tied to emotional ones. Harvey is disfigured and his girlfriend Rachel dies in the accident orchestrated by the Joker.

Once Dent becomes Two-Face, he has a tactic of flipping a coin to decide whether to kill someone. This is exactly like something the Joker would do it’s totally random, a combination of a lighthearted game and a terrible violence. So the Joker succeeds in remaking the idealistic do-gooder Harvey in his own image. At the same time, the idea that the Joker’s own scars are tied to emotional trauma comes only from the stories the Joker tells so this makes that explanation completely suspect. The Joker himself isn’t broken by his scars the way Harvey is. He takes a perverse pleasure in his scars. So it seems overwhelmingly likely the Joker did this to himself and not for love, but to express an identity that he’s proud of. The main thing we know about the Joker’s scars is that he sees them as an asset a psychological element in his violent, fear-inspiring game. The Joker wants to break people, to convince them that their society is illogical and meaningless. And these strange, unexplained scars are his way of proving to people that their world is insane.