The characters for Komeiji literally mean "ancient and pure/bright earth". Koishi is not written in Kanji, but it's supposed to mean "love" or "beloved".
In Koishi's profile, it is said that she met a human at the shrine at the top of the mountain who made her regret closing her third eye and want to learn more about this human. Though it may seem like this was talking about Sanae, once put in to context by the extra stage of Subterranean Animism, it is clear that the human it was talking about is either Reimu or Marisa.
After the fight with the heroine, she starts to become curious about the Human race and begins to investigate them. The Humans can't see her because of her ability to control the Subconsious. (She uses her ability to become invisible to humans).
Koishi's musical theme, Hartmann's Youkai Girl, may refer to Eduard von Hartmann (whose most famous work is entitled The Philosophy of the Unconscious) or Heinz Hartmann (as many of Koishi's spellcards seem to make references to ego psychology.) The track itself is also arranged so that it has symmetrical parts, making it sound remarkably similar to itself even when played reversed, possibly as a reference to the practice of backmasking.
Satori and Koishi's design may have been inspired by the 1939 painting by Kahlo titled "The Two Fridas". The subject is essentially two identical women (sisters/clones/twins...) with visible hearts and external arteries that seem to connect and wrap around their shoulders and arms, where one of them has a cut artery and a pair of scissors in hands (Resembling Koishi, who has closed her third eye).
If her and Satori's color schemes are inverted, they will both somewhat resemble each other.