This is a notification option from apps on the App Store. Unlike remote notifications, these messages do not come from Internet services. 'Application creates notification details right on the phone and switches to the system'. Often this message is used to tell users what the application wants you to do, for example, in-to-do-list or feeding the pet in the game.
Up to this point, all reported cases of errors come from applications downloaded from the App Store and use Apple's Local Notification. Turning on DND doesn't help fix this.
This is a rather unpaid solution but with the least problems: