Is Chicago the "City of Angels"?

No. The nickname "City of Angels" belongs to Los Angeles — derived from its Spanish name El Pueblo de Nuestra Señora la Reina de los Ángeles ("The Town of Our Lady the Queen of Angels").

Chicago's nicknames include "Windy City" (likely from 19th-century boosterism, not weather), "Second City" (post-fire rebuilding / cultural rivalry), and "City of Big Shoulders" (from Carl Sandburg's poem).