The Peter Pan collar is indeed named after the fictional character from J.M. Barrie's book (which also popularized the name 'Wendy'). The 1905 stage production starred Maude Adams as Peter Pan, and the collar on the costume she wore to play "the boy who wouldn't grow up" was thus immortalized, though it was a style worn previously.
The French name for the round collar style, col Claudine, is also derived from literature, named for the title character in Colette's 1900 book 'Claudine à l'école'. The collar has always been associated with youth, being a popular style for infants' clothing as well as for female school uniforms in France.
As a fashion perennial, if not an item in every woman's closet, it imparts a certain youthful and adorable French-ness to the wearer.