No, it is a lowercase shortening of the acronym for Ideas From the Deep, which is what they called themselves before id. Ideas From the Deep becomes IFD becomes ID.
"Taking its name from Freud's primal, instinct-driven face of the human psyche, id Software is, by general acknowledgement, the coolest game shop in the world."
"Taking its name from Freud's primal, instinct-driven face of the human psyche, id Software is, by general acknowledgement, the coolest game shop in the world."