I don't know how Signal works but it is possible to send a silent encrypted push notification that the app can decrypt and show as a local notification.
We use React Native for our apps and React for desktop - I'd say about 95% of the Android/iOS code is shared and we're able to share a lot of business logic between mobile and web. Having all of the code together in one repo has worked really well for us and some of the most complicated stuff is shared between all platforms.
If you want to share UI across platforms you could potentially use something like https://github.com/lelandrichardson/react-primitives to do that, but we don't and I'm not sure we'd want to because we don't want to use the same design on all platforms.