The problem the poem sets
A man leaves for twenty years. His son was an infant and does not know his face. His wife has been told he is dead by a series of liars. His appearance has been altered by hardship and then deliberately disguised by a goddess. What could establish that he is who he says he is?
The second half of the Odyssey is a systematic answer to that question, and the answers are all different.
The recognitions, and their mechanisms
Telemachus (Book 16) — by declaration. No proof at all. Odysseus is transformed by Athena, his son takes him for a god, and he simply says that he is his father. Telemachus initially refuses to believe it. It is the weakest evidence in the poem and it works, because the two have no shared past to draw on.
Argos the dog (Book 17) — by scent and instinct. Immediate, certain, and useless: the dog cannot speak, and dies.
Eurycleia (Book 19) — by the body. She finds the boar-scar on his thigh while washing his feet. The poem then breaks off for some seventy lines to tell how he got the scar, hunting on Parnassus, and how his grandfather named him. It is involuntary recognition, and it is immediately suppressed by force: he takes her by the throat.
Eumaeus and Philoetius (Book 21) — by the scar, shown deliberately. The same evidence, offered rather than discovered, as credentials before the killing.
Penelope (Book 23) — by a shared secret. She orders the marriage bed moved; he explains, angrily, that he built it around a living olive trunk rooted in the ground. The proof is a fact only two living people knew.
Laertes (Book 24) — by the scar and by memory. The scar again, and then the trees: thirteen pear, ten apple, forty fig, named by a man recalling what his father gave him when he was a boy in the garden.
What the sequence argues
The mechanisms are ranked, and the ranking is an argument. Physical marks are the weakest kind of proof, because a body can be counterfeited in principle and because the scar identifies a body rather than a person. The bed and the trees are the strongest, because they are shared history: things that cannot be reported by a third party because nobody else was there.
The poem's implicit position is that identity is constituted by relationships and by what two people know together, not by appearance. This is why Penelope's test is better than Eurycleia's discovery, and why the poem gives it the climactic position.
The Greek terms
Anagnōrisis, recognition — the word Aristotle takes up in the Poetics, where he classifies its kinds and rates recognition by tokens as the least artistic and recognition arising from the action itself as the best. The bed is his kind.
Sēma, sign or token — the scar, the bed, the trees. The word also means a grave-marker, which gives the recognitions a quiet undertow.
The ethical tension
Recognition in this poem is always preceded by testing, and the testing is often cruel. Odysseus watches his wife weep, his nurse grieve, and his father pour dust on his head, in each case while withholding what would stop it. The poem does not apologise for this and does not present it as a flaw to be corrected. It is what a man who has survived by concealment has become.
Later interpretation
Erich Auerbach's Mimesis (1946) opens with the scar passage and uses it to characterise Homeric style: everything brought into the foreground and fully illuminated, with no depths and no perspective. The reading has been contested but it remains the standard entry point for discussion of how the poem narrates.