For IRC bouncers there's many, but I'm not aware of anything that has a web interface, phone app, push notifications and all kept in sync. The market is pretty small.
Alternatively there's many bridges these days so you can use Matrix with one of the many apps supported.
> I'm not aware of anything that has a web interface, phone app, push notifications and all kept in sync.
IRCCloud does. Or if you want an open source stack: Soju as bouncer, Gamja as web interface, and Goguma as Android app. If you have a paid Sourcehut account you get Soju+Gamja hosted for you at https://chat.sr.ht/ .
I'm not sure The Lounge and Quassel support push notifications, but they otherwise fit your requirements.
Alternatively there's many bridges these days so you can use Matrix with one of the many apps supported.