System Information
Operating system: macOS-11.6-x86_64-i386-64bit 64 Bits
Graphics card: NVIDIA GeForce GT 750M OpenGL Engine NVIDIA Corporation 4.1 NVIDIA-16.0.13

Blender Version
Broken: version: 2.93.4, branch: master, commit date: 2021-08-31 09:23, hash: rBb7205031cec4
Worked: (newest version of Blender that worked as expected)

Short description of error

If two Library overridden armatures that have been duplicated from the same link are in pose mode at the same time, when you select one bone, it will appear to select the same bone on the other duplicate and the pivot point will adjust to be between the two as well. In this case, when you move it, it will still only move the one that has actually been selected.

To take it further, if you shift select a bone from the other armature, it will it will only affect both bones on the second armature, so now the incorrect bones are being affected.

Exact steps for others to reproduce the error

1. Create armature with at least 2-3 bones that are not connected to each other (for better visibility of the issue) and save the file.
2. Create a new file, link in the armature and make a library override.
3. Right click on the your object in the outliner and Duplicate Collection. You could also manually link in a second version of the same collection and this will produce the same result.
4. Slide the duplicate over for better visibility.
5. Shift select both objects and switch to pose mode.
6. Select on a bone on one of the duplicates and scale, move or rotate it. At this point you should see that the same bones appears to be selected on the other object, but only the one you actually selected it moving. The pivot point is between the two bones.
7. Now shift select a bone on the second object and only the two bones on the second object move.
8. If you have one bone selected and you shift select the same bone on the second armature, only the bone on the second armature will move.

Let me know if you need any more information. Thank you!

Zip file contains a .blend file with the original armature called 'test_armature' and a .blend file with the duplicated library overrides called 'library_override_bug_test'