4 ideas
Trapped in Polyphemus's cave, Odysseus gives his name as 'Nobody,' so when the blinded Cyclops cries that 'Nobody is killing me' his neighbors do not come to help, and the men escape strapped under the sheep. Then, sailing away, Odysseus shouts his real name and lineage back at the shore, and Polyphemus prays to his father Poseidon for revenge.
Odysseus comes back to Ithaca dressed as a beggar and watches how each person treats a stranger who seems to have nothing. People show their real allegiance only when they think it earns them nothing: the swineherd Eumaeus shares his food and the old dog Argos knows him, while the suitors and the disloyal servants throw insults and footstools. Hiding his identity lets him sort friend from enemy before he acts.
Real identity is proven by a private detail that an impostor could not know or fake, not by what the person claims. The nurse Eurycleia knows Odysseus by the boar-hunt scar on his leg.
The epic treats xenia, the sacred rule of hospitality between host and guest, as the moral law that decides who deserves to die. The suitors break it by eating up the absent host's household and abusing a stranger at his table, and so the slaughter in the hall is shown as justice, with the gods' approval, rather than as excess.