Nevermind, just got into blender manual, seems like armature instances are not supported for fbx export.
If anyone wants to know more go here:
Export all objects of a certain collection
Pretty much straight forward. Just pass use_active_collection=True argument when calling export_scene.fbx() operator and assemble your filepath using Collection.name:
C = bpy.context
It is because you didn't apply rotation and scale.
So select that object (just the part which flies away) and press CTRL-A and then "rotation & scale".
Then export and import again -> works ...und Oma kann auch wieder besser sitzen ;)