Hi Ali,
Yes, duplicating an Object (Linked or not) makes a new Object, so the Random Value of the Object Info Node will be different for each ‘copy’. That is not you, doing anything wrong.
The problem with your looping, might be, because your rings flicker too slowly, or not at all, which would create a visible jump between the first and last Frame…
Also, make sure you are using Linear Interpolation between the Keyframe (for the same reason).
