But... I didn't install any "battery manager" app
You didn't, but most new phones nowadays are bundled with some kind of battery/power saving manager software that should be set up in order to allow apps to show notifications. Each manufacturer has a different battery manager app.
Are there any other apps that prevent Lady Pill Reminder from working properly?
Yes: any app that "force close" Lady Pill Reminder (task killers, antivirus, task managers, app cleaners...). So you have to set them up correctly to allow Lady Pill Reminder showing notifications.