The encounter with the runaway works to show that the narrator identifies with this man just as he earlier identified with nature, with the clam-diggers and with the marriage party. The "self" in Whitman's long poem is defined by the vast array of concepts and people that populate the world...
of the poem.
The “I” of the poem is quite clearly, then, not the everyday self, the small, personal ego that is unique and different from all other selves.
The self is defined by association, by acceptance, and by identification. The narrator sees himself in his world. Effectively, this allows for a definition of self that is unbounded, universal; expansive.
We can see this idea in a great majority of the different sections of the poem. In section 10, we are given a sense of the qualities of character that go into defining the self.
While the narrator is pleased to accept the hospitality of others as he does with the boatmen, he is also willing to give hospitality to another. He accepts and conveys.
The boatmen and clam-diggers arose early and stopt for me, I tuck'd my trowser-ends in my boots and went and had a good time; You should have been with us that day round the chowder-kettle.
With the runaway, the narrator demonstrates his compassion, which is a necessary aspect of his character. Without empathy and compassion, the expansive identity created in the poem would be either impossible or lacking integrity.
However, the narrator possesses this trait in good measure. He is willing to take in the runaway in his weakness, to give him strength and, finally, protection.
He staid with me a week before he was recuperated and pass'd north, I had him sit next me at table, my fire-lock lean'd in the corner.
The narrator's relationship with the runaway can be described in these ways: a relationship of identification and an example of the qualities of the narrator's character.