FWIW check your doorbell doesn't have a way to change its ID. Probably some jumpers or DIP switches or something in the battery compartment of both parts.
Ours is extremely homebrew, and is part of a complex interconnected system with a squillion other non-doorbell features[1] (and at least one major bug I haven't been able to work around satisfactorily), represents a huge number of Kim-hours of work, a four-figure hardware cost, and wouldn't be much use unless you're already heavily invested in IRC[2] as a user interface.
But more generally, this is the sort of thing Arduinos are good at. There seem to be various solutions to generating push notifications from an internet-connected Arduino, although the first three I clicked on seem to involve some third-party service that I've never heard of. You could always do something standards-based like send an email, or low-tech and real time like a phone call[3].
[1] Everything from a gentle message if the computer thinks someone has fallen asleep in the bath, to re-enacting the Nostromo's self-destruct sequence when the fire alarm goes off, via telling you what it was you came upstairs for.
[2] Mainly this shows the age of our original system, but the great thing about having missed out on the cloud computing IoT revolution is that it all still works without an internet connection. You can still just about get an IRC client for smartphones with a working highlight function.
[3] Back in the days when SMSes could take a several of minutes to be delivered, I used to have a Nokia wired to the Fiat Of The Apocalypse's alarm system via a PIC. It would generate the appropriate AT command string to attempt to make a HSCSD modem call to my phone when the alarm went off (at which point I'd recognise the number and immediately reject the call). I thought this was pretty clever until I saw Postman Piers's version based on a relay shorting one of the speed-dial buttons on a much older and less impressive Nokia.
