In Black 2 and White 2, Ghetsis appears to have become somewhat
insane, going so far as to command Kyurem to impale the player with
icicles at one point (although failing due to N's intervention). Also
he is visibly mentally unstable, bursting into fits of rage that
prompt him to shout and to aggressively slam his cane to the ground. A
Shadow Triad member points out Ghetsis loses control, which further
evidences this behaviour to be recurrent.
Yes, the cane will stay there for the rest of your BW2 life.
And in the picture on the right, that is the cane close up.