Hubitat and SmartThings notifications work differently; with SmartThings, it's tied to your account and will notify an app/device logged into that account. With Hubitat, notifications go to a device, so they're just like regular device commands (e.g., turning on a switch, where you have to select a device to turn on). But, when you log into the Hubitat mobile app on a new device, you should be prompted whether you want to create a new mobile app device or re-use an existing mobile app device. If one device is replacing the other, it sounds like you might want to use the second option instead of the first--then, the existing device will be associated with your new app/device, and you won't need to swap anything out.
If you do want to create a new device on Hubitat for some reason, then there is at least something that can help you without needing to manually swap it in all apps/rules: the Settings > Swap Apps Device feature, added in platform 2.3.1.
I also have a custom app/driver that can be used to group or "proxy" notifications through a virtual device, then you'd only need to swap it out in that one app and will affect all, but with both of the above, I'm not sure you'd really need that.