The decision to call Reilly's masked crimefighting persona "The Spider" instead of "Spider-Man" was made as a callback to early heroes of the 1930s like The Shadow.

Spider-Noir (Part 2)

"Ben Reilly, a seasoned, down-on-his-luck private investigator in 1930s New York, is forced to grapple with his past life, following a deeply personal tragedy, as the city's one and only superhero." (Synopsis via IMDB)