Almost certainly made up for the game
Celebrimbor made three Rings of Power for the Elves (the three Elven Rings worn by Gandalf, Elrond and Galadriel at the time of the War of the Ring). However none of them was a second "One Ring to rule them all".
In those days the smiths of Ost-in-Edhil surpassed all that they had contrived before; and they took thought, and they made Rings of Power. But Sauron guided their labours, and he was aware of all that they did;
The Silmarillion: Of the Rings of Power and the Third Age
Page 287 (George Allen and Unwin 1977 hardback edition)
It's hard to prove a negative, but there is certainly nothing in the Appendix of The Lord of the Rings about a second One Ring. I don't recall reading any reference to such a thing in The Letters, or any of the books published by Christopher Tolkien.
If such a Ring had existed, Elrond would surely know about it, and it would have been discussed at the Council of Elrond. In fact this is all Elrond says about the forging of the Rings:
Then all listened while Elrond in his clear voice spoke of Sauron and the Rings of Power, and their forging in the Second Age of the world long ago. A part of his tale was known to some there, but the full tale to none, and many eyes were turned to Elrond in fear and wonder as he told of the Elven-smiths of Eregion and their friendship with Moria, and their eagerness for knowledge, by which Sauron ensnared them. For in that time he was not yet evil to behold, and they received his aid and grew mighty in craft, whereas he learned all their secrets, and betrayed them, and forged secretly in the Mountain of Fire the One Ring to be their master. But Celebrimbor was aware of him, and hid the Three which he had made; and there was war, and the land was laid waste, and the gate of Moria was shut.
The Lord of the Rings Book Two, Chapter 2: The Council of Elrond
Page 242 (Single volume 50th Anniversary Edition)
I have to conclude that this was made up only for the game.