I see that scene as a visible manifestation of <PERSON>'s identity crisis.
<PERSON> was born to be <PERSON>, son of <PERSON>, son of <PERSON>, King under the Mountain, but this was denied by the coming of <PERSON> and the sack of Erebor.
At the battle of Azanulbizar at the eastern gate of Moria he earns himself the name of <PERSON> thanks to his duel against <PERSON>, which turned the tide of the battle and gained the dwarves a victory, though a bitter one because of their heavy losses.
At that point, as he strongly believes his father to be still alive, <PERSON> does not consider himself a king - not even a king in exile -, even if that's actually how the other dwarves see him:
[<PERSON>:] “That is when I saw him: a young dwarf prince facing down <PERSON>.”
[...]
[<PERSON>:] And I thought to myself then, there is one who I could follow. There is one I could call King.”
(from AUJ)
[Dwalin] “You were always my king. You used to know that once.”
(from BOTFA)
From his own point of view he is <PERSON>, the one who defeated <PERSON>. The fact that he kept the oaken branch, carving it into a proper shield and adding metal spikes, is a visual clue of how much he clings to that title, valuing that duel more than anything else he accomplished.
Note that, once again, the other dwarves think otherwise:
[Balin:] “You don’t have to do this. You have a choice. You’ve done honorably by our people. You have built a new life for us in the Blue Mountains, a life of peace and plenty. A life that is worth more than all the gold in Erebor.”
(from AUJ)
In goblin town, the Great Goblin malevolently rubs salt into the wound:
[Great Goblin:] Well, well, well, look who it is. 381
<PERSON> son of <PERSON>, son of <PERSON>; King under the Mountain.
Oh, but I’m forgetting, you don’t have a mountain. And you’re not a king. Which makes you nobody, really.
But <PERSON> is far more shocked to learn about <PERSON>:
[Great Goblin:] I know someone who would pay a pretty price for your head. Just the head, nothing attached. Perhaps you know of whom I speak, an old enemy of yours. <PERSON> astride a White Warg.
[Thorin:] “<PERSON> the Defiler was destroyed. He was slain in battle long ago.”
[Great Goblin:] “So you think his defiling days are done, do you?”
And finally, the scene in question: <PERSON> finds out that <PERSON> is still alive, and stronger than him.
His most valued deed is worthless, his battle title means nothing. He will later say <PERSON>:
[<PERSON>:] “Do not speak to me as if I was some lowly dwarf lord…”
“As-As if I were still...<PERSON>...<PERSON>.”
“I AM YOUR KING!”
(from BOTFA)
His losing the physical oaken shield parallels his losing the identity of <PERSON> - the war hero - after having already lost his kingship. Now he must desperately fulfil his quest to reclaim Erebor, in order to prove his value.
Sources:
* AUJ fan transcript
* DOS fan transcript
* BOTFA fan transcript 381