I see that scene as a visible manifestation of Thorin's identity crisis.
Thorin was born to be Thorin, son of Thrain, son of Thror, King under the Mountain, but this was denied by the coming of Smaug and the sack of Erebor.
At the battle of Azanulbizar at the eastern gate of Moria he earns himself the name of Thorin Oakenshield thanks to his duel against Azog, 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, Thorin does not consider himself a king - not even a king in exile -, even if that's actually how the other dwarves see him:
[Balin:] “That is when I saw him: a young dwarf prince facing down the Pale Orc.”
[...]
[Balin:] 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 Thorin Oakenshield, the one who defeated Azog the Defiler. 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. Thorin son of Thrain, son of Thror; 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 Thorin is far more shocked to learn about Azog:
[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. A Pale Orc astride a White Warg.
[Thorin:] “Azog 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: Thorin finds out that Azog is still alive, and stronger than him.
His most valued deed is worthless, his battle title means nothing. He will later say Dwalin:
[Thorin:] “Do not speak to me as if I was some lowly dwarf lord…”
“As-As if I were still...Thorin...Oakenshield.”
“I AM YOUR KING!”
(from BOTFA)
His losing the physical oaken shield parallels his losing the identity of Oakenshield - 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.
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