How does Odysseus search for his identity in the Odyssey?
Odysseus was never in a position to doubt his identity, as if he had a bout with amnesia in a shipwreck. Therefore, he does not seek to find out who he is. A better way to look at it is he seeks to reestablish his identity.
He is seeking to reestablish his identity because he has been fighting in the Trojan War. He has been away for a long time and on his way back, he was shipwrecked and held prisoner by Calypso. Also he faced many obstacles on his voyage home, which prolonged his time away.
In the meantime, suitors on his island were seeking to win the hand of his wife, Penelope. They all assumed that he died and wanted his wife, possessions, and status. So, Odysseus's homecoming was also a way of asserting his past identity. He was alive and the rightful leader of his home, Ithaca.
Odysseus seeks his identity in several ways.
The first is that in his literal search—for a way home—he is also searching for himself. He’s been away from home for years. He’s been out having adventures, but he’s also been places where people don’t know him, or where they meet him for the first time. At home, his wife and old dog, who knew him before his travels, wait for him.
A second search is for Penelope. Yes, that’s another person, but she’s his wife, and the great love of his life. She is part of him, and so in finding his way back to her, he’s seeking himself.
A third is seeking Telemachus. In this he seeks a new identity. He’s been gone for so long that he hasn’t had a chance to be a father. Now he comes home to a new role.