Thoose kinds of rigs use a lot of Blender specific operators that cannot be stored into a generic format like Fbx.
If you want to import your rigged model into a new .blend file this is the suggested procedure:
open the Source file; if not present, create a new collection and move everything related to the rig and the model into this new collection. Name it in a recognizable way, save and close.
open the destination file or a new file, go to Header Menu > File > Append, navigate your hard drive to the source file, double click on it, double click the "Collection" folder, double click on the newly created collection. All subcollections contained in the main collection will be imported.
Be aware that the Append operator will import everything that it's connected to the rig and/or the model by parentship, constraints, assignements, ... even if they aren't placed in the choosen collection.
The "Link" operator is similar to the "Append" one, but it's more complex, and it's useful when dealing with larger projects and teams, when different workstations with different people work on the same project, search on manual and tutorials for more details.
Use Fbx only to import/export from Blender to others applications, knowing that some informations and structures will be lost in translation.
