Celeborn is not a Ringbearer, or at least he does not hold one of the Three Rings of the Elves; he might possess one of the lesser rings that were crafted by the Elves of Eregion.
Tolkien himself waffled on the subject of Celeborn's history. In The Lord of the Rings he is said to be a Sindar Elf of Doriath who had never been to Aman. He and Galadriel met in Beleriand and were wed in Doriath. In Unfinished Tales we learn of two alternate histories: Tolkien's original conception of Celeborn as seemingly a Nandorin Elf who had always dwelt east of the Misty Mountains; and a later version where Celeborn was a Telerin Elf of Aman with the Quenya name of Teleporno (see Unfinished Tales, "The History of Galadriel and Celeborn").