He is, and he knows it. Throughout the book he references a “terrible purpose” that has been thrust upon him, which is the slaughtering of trillions (and may also refer to the Golden Path, which is mentioned more in Children of Dune and later books). The in-universe translation for Kwisatz Haderach is “Shortening of the Way.” The BG basically wanted a male Reverend Mother who could bridge the gap between the shared memories of the BG (the past), the present, and the future using the KH’s prescience. Paul wasn’t supposed to be the KH, nor was he supposed to be a male. Jessica was instructed by her BG teachers to produce a girl, but she fell in love with Duke Leto (she wasn’t supposed to), and gave him a boy (BG can control these things). The theory is they were likely looking to control the universe with this One who could see everything with their prescience. Paul could, and carried out his great jihad to control the empire, but not for the BG. The BG also fucked themselves over by planting the prophecy amongst the Fremen thousands of years before the events of Dune, which heralded the arrival of Paul and his mother, which the Fremen believed and allowed Paul to rise to power.