Is there any validity to the statement that sincere repentance requires abstaining from sin for forty days, and must these days be continuous?

No evidence has been found to support what you mentioned. The door of repentance is open to every sincere and devoted repentant, for Allah has promised mercy and forgiveness to whoever repents and seeks forgiveness after committing a sin, as affirmed by His saying, the Almighty: "Your Lord has decreed upon Himself mercy: that any of you who does evil in ignorance and then repents after that and corrects himself - indeed, He is Forgiving and Merciful" (Al-An'am: 54), and "And whoever does a wrong or wrongs himself but then seeks forgiveness of Allah will find Allah Forgiving and Merciful" (An-Nisa: 110), and other verses and hadiths that emphasize the acceptance of repentance as long as the death rattle has not reached the throat, and that Allah extends His hand by night and day to accept the repentance of the sinner. To remain steadfast in repentance, one should frequently read books on spiritual matters and texts of encouragement and deterrence, associate with the righteous, and avoid environments of sin and their people, as this strengthens repentance.