Keep in mind that all Thread devices are not Matter devices. Thread is just for a device’s connectivity, analogous to WiFi. Matter can be used for device control. Some, but not all, Thread Border Routers are also Matter Controllers, so while they “speak” Matter they don’t have any QR to join another fabric as they form a new fabric themselves.
The HA screen you captured shows any TBRs that have been detected on your network. If HA has saved the network’s security credentials, it displays the circled (i). The NEST network has no border routers anymore but must have been detected and its credentials saved in HA previously. If you no longer have that network or need those credentials, you can delete them. Unfortunately, they may still be saved on your phone e.g. via Google Play services.
If you want your Google streamer to join your OTBR thread mesh, it needs to have the same credentials as that mesh. Unfortunately I’m not aware of any UI that permits you to set or change the Thread credentials of a Google TBR. It kind of just picks them for you when you set it up. Sometimes, if the phone you’re using for setup already has credentials saved via the Google Play services, it will pick those.
So one approach could be to erase the Google Play data, sync the OTBR credentials to your phone so they’re the only ones there, factory reset the streamer, and use that same synced phone to set up the streamer again, hoping it pulls the HA credentials instead of creating new ones. If you try this and it works, let us know!
In general the tools for Thread management are kind of a mess right now, at least from a UX perspective, in particular if trying to combine different vendors. Since Google (and others) don’t let you set credentials, it’s often easier to set your OTBR to use their credentials, rather than the other way around. Hopefully these UIs improve.
All that said, you mentioned wanting to extend your network, and adding a second TBR is not really the best way to go about that — Thread repeater (router) nodes are preferable, similar to how Zigbee and Z-wave work. Adding TBRs does add some resilience, but too many can actually make things worse. You can even make your own repeater Thread node using a esp32-h2 for around $5.
