So, Entertainment District isn't really a perfect translation of what that area is, its actually closer to "Red Light District", I'm guessing they changed it for western sensibilities. The district is where prostitutes and courtesans work, the three houses that they infiltrated are places where these girls are trained since youth to do this kind of work. The reason Uzui said Zenitsu was too young to approach the women in the balcony was because they were prostitutes.