It's a myth in the most literal way. Fleming published and promoted his results despite a lack of reproducibility. By the time he won the Nobel Prize, he had backformed or misremembered a folksy story about an open window. That's textbook mythmaking.
It can both be fine to have a glib story to tell schoolkids and important to recognize that the actual intellectual process is messier and more complex.
I have now actually read Flemming's 1929 manuscript that first described penicillin [0]. It is a careful and well documented scientific report describing the action of penicillin on various species of bacteria, how to produce it, and some of its chemical properties. It describes how penicillin can kill bacteria isolated from the throats of nurses, and shows that it has low toxicity in mice, and is possibly safe for use in humans: "Constant irrigation of large infected surfaces in man was not accompanied by any toxic symptoms, while irrigation of the human conjunctiva every hour for a day had no irritant effect."
It is far from having a "lack of reproducibility" and in fact allowed others to quickly and accurately replicate his discovery.
The path to his discovery may have been difficult to replicate, but the fact that the mold could kill other bacteria was not, and was immediately replicated.
It just wasn't seen as relevant because, at the time, few people imagined its internal use in humans and it was instead seen more as a tool for other microbiologists and the like. The jump to "And then I see the disinfectant, where it knocks it out in a minute. One minute. And is there a way we can do something like that, by injection inside or almost a cleaning?" took quite some time.
It can both be fine to have a glib story to tell schoolkids and important to recognize that the actual intellectual process is messier and more complex.