If for whatever reason this is not possible (e.g. because it clashes with Event FX), then at least it would be nice to provide a way to select all shared copies of the currently selected item. Then, when all items are selected, they can easily be renamed and coloured simultaneously via the context menu. However, I much prefer the new name to propagate automatically, it's just less work for the user.
The reason I mention Event FX is that two shared copies of one event can have different Event FX on them, and in that case it would make sense to name them differently, or even color them differently, even though the MIDI data is the same. There are other settings that also can be applied to each shared copy individually, for example, Transpose. In my opinion this is crazy. Shared copies by definition should represent the exact same sound, so adding Event FX to a shared copy should add it to all shared copies, and renaming, transposing, etc should also be applied to all shared copies. If the user needs to add an FX to only one of the shared copies, the user should use the Separate Shared Copies command, and then apply the FX.
The current implementation makes shared copies very difficult to manage, because it's too easy to end up with many shared copies all named differently, and then accidents begin to happen throughout a project.