Actually it doesn't seem possible.
When you open the Add menu (Shift+A) you will see a list of all currently loaded Clips that have been opened in a
Movie Clip Editor respectively their current Name properties.
Once you Add a "Clip" to the Sequencer it will initially get the original MovieClips name. But the corresponding MovieClip from the Strip of the Type MovieClipSequence being titled Clip in the Strips properties is only internally referenced. You cannot access it from the Sequencer anymore nor through the Python API. You can proof this via an Outliner view which is set to
Datablocks Display Mode. There you can see all your
Movie Clips in your Blendfile Data. Those have a name Property which can be set through Python via
bpy.data.movieclips["clip_name"].name. Not that the Movie Clip is of the RNA Type
But when you look at your
All sequences you can see all of your added Strips including the added Movie Clip (with it's original name) and an own name property which can be edited via
bpy.data.scenes['Scene'].sequence_editor.sequences_all["clip_name"].name. Note its RNA Type
When you now change the name of the MovieClip in the Sequencer you change the name of the MovieClipSequence object which can differ from the name of your MovieClip object. And only when they match the Strip in the sequencer will only show one name followed by a
| and the MovieClipSequencer objects frame length. But as soon as you change one of these names, both will be displayed, the MovieClipSequenc objects first, followed by a colon
: and the MovieClip objects name (followed by the
| and the MovieClipSequence objects frame length). Unselected it's all a bit hard to read as imported Clips get a dark blue strip color by default.
But as soon as you see a colon
: in a Clip strips name in the sequencer you can no longer match a MovieClip and a MovieClipSequencer as you cannot get the other ones name from one of these objects in Python as you can see there is no such link neither in the Python autocompletion nor in the Outliners Datablocks display mode. It's all handled internally by Blender for now.