It should be mostly possible with ntfy, except everything would have to use it, and most apps just use one of the centralized push notification services.
I have a mildly smart home, and the server sends a push notification via ntfy. I have one of those mobile automaton tools on my phone that plays an mp3 when it sees that kind of notification. Then I have audio notifications turned off in the ntfy app my phone. With sufficiently advanced rules, I think you could set up something like you suggest; if not, a custom app could do it - one notification, and then there rest are silent until you dismiss it.
It's not a complete solution (unless you use Matrix, which can be configured to send notifications via ntfy), but my point is there's little technically preventing a solution being hacked together with existing tools. Except for those apps that only use Google or Apples push notification systems. Which is most of them.