Nothing's thing was a branded version of Sunbird
https://www.sunbirdapp.com/ which is technically similar to the iMessage part of Beeper Cloud. It is essentially a way for you to login with your Apple ID on a Mac Mini in a server farm, and then interact with its desktop iMessage client from an Android app.
This was done because Apple obfuscates how its notification system works, so the cheapest short-term solution is to just use real Apple hardware.
When Nothing released it, it was found to have many flaws, which is where that video comes in. Nothing unreleased it and hasn't followed up since.
Beeper Mini uses the long-term solution of reverse engineering Apple's notification system so that it can run independently of an Apple device.