The Lion Guard has been slowly expanding on The Lion King’s backstory for a while now. And they just revealed how Scar got his well, Scar.
So when Scar died in the movie instead of going up to the sky which symbolizes lion heaven, he was instead sent to a nearby volcano. In a musical number he explains to a snake how he got his scar while he was on duty running the Lion Guard.
So one day while he was patrolling he ran into another lion who wanted to overthrow Mufasa. Sound familiar? He was curious but also knew the lion was up to something. Promising that they would rule the land together after he became king Scar followed him to an undisclosed location. Which in people speak means RED FLAG. So when he gets up the mountain he is attacked by a venomous snake across the eye. Which is how he got his scar.
The lion he met then tells him that he can suck out the poison only if he helps overthrow his brother. Scar refuses and kills both that lion and the snake.
The nickname came about when Scar saw his brother Mufasa and he flippantly called him Scar and walked away. Either not caring enough about him or calling him a name after what Scar had done sent him over the edge. And the plot to overthrow the throne for himself was born.