There are a few reasons the title is called "The Secret Sharer." The narrator is given his first command of a ship. He wonders if he will live up to the standards he has set for himself in his mind. " . . . I wondered how far I should turn out faithful to that ideal conception of one's own personality every man sets up for himself secretly." He feels like a bit of an outcast relative to the rest of the crew. So, initially this ideal conception of himself is only in his min: a secret he keeps with himself.
His real test of living up to his idealization of himself occurs when he meets Leggatt. He decides to hide Leggatt even though Leggatt is on the run (swim) for killing a man. The captain develops a kinship with Leggatt, even calling him his "double." The captain does whatever he can to hide and protect Leggatt. He keeps Leggatt and his crime a secret from his own crew and lies to protect him from Leggatt's commander. The captain and Leggatt "share" the secret of Leggatt's crime and being a stowaway on the captain's ship. They also share this double kinship, a bond that neither of them can share with anyone else. And while the captain does share this bond with Leggatt, it is never made explicitly clear (to the reader) why he chooses to help Leggatt even if it means endangering his own crew. Thus, part of the significance of their bond is a secret that even the reader of the story is not privy to.