I used to make similar mistakes years ago, for a long time thinking it was "Eye Dee Software", so I'm posting this to enlighten others, rather than being a grammar commie.
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."
I've been calling them eye dee software since the first wolfenstein came out. I forget when I learned it rhymed with lid, but I just can't make myself say it that way.