"As for the Master he saw there was nothing else for it but to obey the general clamourm, for the moment at any rate, and to pretend to believe that Thorin was what he said."
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The Master did not want to anger the Elven-king and didn't really believe Thorin; he simply went along with the clamour because of the excitement and belief of the people. The matter of Thorin's identity was settled by the people rather than the Master.
"Then the Master hesitated and looked from one to the other. The Elvenking was very powerful in those parts and the Master wished for no enmity with him, nor did he think much of old songs, giving his mind to .trade and tolls, to cargoes and gold, to which habit he owed his position. Others were of different mind, however, and quickly the matter was settled without him. The news had spread from the doors of the hall like fire through all the town. People were shouting inside the hall and outside it. The quays were thronged with hurrying feet. Some began to sing snatches of old songs concerning the return of the King under the Mountain; that it was Thror's grandson not Thror himself that had come back did not bother them at all. Others took up the song and it rolled loud and high over the lake."
The Hobbit/ Chapter 10