Similarly to this answer about syncing Do Not Disturb mode across devices, this answer appears outdated as well.
You can now, with Apple's Focus Feature, allow certain applications or certain users to create and trigger notifications that will display and sound for you in your chosen focus. This has to obviously be configured in settings but is absolutely possible.
Here is the official Apple Information as of September, 2022. It will naturally go out of date in a few years when all these tech companies feel like slapping another new coat of paint on top of all of their old work.
Set up your Focus. You can select allowed or silenced notifications from people and apps, connect your Lock Screen or Home Screen, have this Focus turn on automatically, and add Focus filters. 0