Rider was actually summoned by Sakura. Due to Sakura’s unwillingness to participate as a Master, Shinji Matou is instead given control over Rider.
Rider’s identity is Medusa, the Gorgon from Greek mythology. Rather than being a regular Heroic Spirit, she is an existence closer to a Divine Spirit, but her actions have made it better to categorize her as an “anti-hero closer to a monster.” With her older sisters Stheno and Euryale, they made up the three Gorgon Sisters who inhabited the Shapeless Isle. She was known as the “woman who dominates,” an anti-hero who became as such due to the dark thoughts of humans. Differing from someone like Hassan-i-Sabbah who was a hero who actual existed, she is said to be one of legends born by the gathering of belief.
The legend of Medusa is different than her actual life. She was said to be a half-human, half-god female beast often portrayed with the form of a snake. Originally she was an Earth Goddess. While all originally beautiful women, she and her sisters were hated by Poseidon’s wife because of his love for them. Athena, jealous of their beautiful hair, transformed them into demons. Secluding themselves in a lightless temple on the Shapeless Isle, they only attacked humans with a reason. Eventually, as a victim of the gods’ unreasonable actions, she was to have her head cut off on their whim. Defeated by Perseus by reflecting her petrification back at her with his mirror-like shield, Pegasus, the son of Poseidon, was said to have sprung from the blood of her headless neck.
In her backstory, She was teased by her sisters for being mortal unlike them, and because Athena sent her followers to attack them, she turned into a man-eating monster that devoured them. At the time of her death, she reminisced that they did actually love one another and accepted she misses her time with them. One of the reasons she is so sympathetic to Sakura is because she is so broken too.