The issue is to make a linked copy of scene.
When you do that, you link the objects of one scene in the other one.
So, that is the same object in both scenes. It is normal, it has same animation in both scenes.
From there, you have to create another collection in original scene, and move armature and character to it.
They should disappear from the other scene.
Relink those objects to copy of originalscene.
Create another collection in copy of original scene. Move them to this collection without equivalent in original scene.
The Object properties tab should show that those objects have 2 users, 2 different collections.
Click on the visible 2 for armature and character or use Make Single User operator on selection of those objects.
The objects are now, different per collection, per scene. They can handle a different animation.
A Library Override is for another purpose. It has to be used when an asset is created as part of a library.
You create character, rig it and mark the collection as an asset and save it as a file, in a folder used as your asset library.
Then, when you create file of the animation ; you link asset by importing character collection (mesh + armature) through asset browser or through File menu > Link > file browser window.
That is a link of data between files, not a link of objects between scenes.
Then, linked asset is fixed as in its original file. There, you have to create a library override to be able to pose armature.
To do it, use the Object menu > Library Override > Make, in 3D Viewport, or use Right Click menu > Library Override > Make, on item in Outliner.
If you want several similar assets in same file, it may be simpler to relink them and recreate a new library override, each time.
Trying to duplicate linked asset may be problematic according to how are set-up your duplicate user preferences.
Maybe it is simpler to start to test library override by creating environment props. Just create a library override to place it.
When it is understood, you can try to create pose assets for animation. Pose libraries are also made of assets to append.
That should allow you, to create animations faster.