According to the Dragon Ball Wikia, this transformation of Kale is called "Super Saiyan Green". Is this an official name or it was made up by fans?
Super Saiyan Green was never official.
The transformation Broly makes is called Legendary Super Saiyan, which has some background since that's what it was called in the movies featuring Broly.
Since Kale resembles that transformation, it's better to call it that, but if I recall correctly, no official name was given yet. Or it's Berserk Super Saiyan.