The Movie That Inspired The Show
Sidney Sheldon, a producer of the show, was a huge fan of a movie called The Brass Bottle. He even ended up using it as the basis of the show! The movie starred Barbara Eden, Burle Ives, and Tony Randall. He was a fan of the female genie concept and used it as the premise of I Dream Of Jeannie. As you can see, it was not a complete Bewitched rip-off.
A Reference In The Monkees
The rock band The Monkees had a TV show of their own, quite similar to Hard Day’s Night by the Beatles. Their show was broadcasted on the same network as I Dream of Jeannie. This explains all the references about the show that appeared in The Monkees. If you need an example, there was a time when one of them finds a genie lamp. Once Davy Jones finds it, he says, “Imagine that — wrong show.”