Two more links for those who like to get REALLY deep into the technical details…
Nordic Semiconductor uses the term “operational network” synonymously with “fabric” and Cisco Networks sometimes uses “virtual network” and both mean the same thing: the set of devices which are allowed to exchange messages based on their IP addresses, even if they are on different physical networks.
When you add a new Matter device to your account, you are “commissioning it“ to the specific matter fabric controlled by that Administrator. But in order to simplify adding new devices, the matter specification also allows that administrator to add the device to the appropriate physical Network (WiFi or Thread at this time) so that the end-user doesn’t have to first add it to the Thread/WiFi network and then add it to the matter fabric. It’s two separate steps, one at the physical layer, and one at the application layer, but the end user only has to make one request, and the administrator takes care of everything.
The following article from Nordic semiconductor details how this two-step process works behind the scenes:
So the device gets added to a physical Network, and then gets added to the virtual fabric, but this might all look like one request to the end-user.
And then the following article from the same source explains what happens behind the scenes if you want to add one device to two different fabrics, for example, to google home and to home assistant.
Again, these are highly technical articles, and most people won’t be interested in this level of detail, but it does explain why it’s so easy to get confused.
Your new Thread with Matter device will be added to a physical thread Network (which might have multiple thread border routers) and it will be added to a virtual Matter fabric.
Then, when you go to add it to a second platform, it will be added to a different virtual fabric, but it stays on the same physical Thread Network.
From a network engineering standpoint, the interesting thing is that the two fabrics don’t have to use the same security codes. It’s just that the end device becomes a trusted member of each of the fabrics.