I haven't heard of an app or a way to do this. The question I see more of is, how I get the phone to notify my when I have the app open? Not sure about other messaging apps but Google Messages I guess won't sound if the app is open, I'm assuming I'm the forgetting. Not sure if that gives you any ideas or not.
The only thing I can think of to get close to what you're looking for would be Bixby, maybe, I don't use routines enough to know everything possible but, maybe you could have a routine that reduces the volume after receiving a message for like 5 minutes or something. I'm pretty sure this is possible with Tasker but again it would be the same thing. The profile would probably be something like, after message received, reduce volume to (some%), wait (some minutes), return volume to previous. Of this wouldn't just reduce Georgia l George's volume but all of them and if the convo went beyond the time you'd get another normal followed by your profile.
I guess it would depend on what you're doing in this interim time, I mean are you usually on YouTube or Facebook or the same something a lot of the time? Tasker or even Bixby might be able to reduce the volume of notifications when you have this app open. Obviously you'd have the phone in hand if the app was open so would it matter if one or all notifications were quieter because you're already paying attention, right?
Perhaps other will have better ideas. This was what I got off the cuff.