Additionally they support "Critical Alerts" on iOS for when you really need to get a notification regardless of whether or not you've muted your device. Probably something similar is possible on Android as well?
Yes, emergency priority: ignores do-not-disturb, full volume, and repeats every x minutes until acknowledged (x is configurable when sending the notification).
On Android, Pushover is able to (optionally) play notification sounds for high-priority messages through the alarm channel like an alarm app would, so it can bypass the device's mute switch.
This has been requested in ntfy as well. I'll likely implement that eventually. Sounds like a great feature.