We may be reading the book from the position of some outside observer, but this third person narrator is definitely in Manjiro's head and looking at things through Manjiro's eyes, which is why we're calling this narrator limited omniscient.
Take this small passage from the scene in which Manjiro has his first private talk with Captain Whitfield:
Manjiro stared at the captain. He had never imagined that a barbarian could appreciate poetry. Or play music. Or express kindness. (2.6.46)
Okay, so any old third person narrator could tell us that Manjiro stares at Captain Whitfield—no special brain access required. But when the word barbarian is busted out, we're clearly glimpsing inside Manjiro's perspective—that's a word he used for white Americans, and the prejudices he holds about what a "barbarian" is capable of come through, too. Our narrator, then, has seamlessly stepped into Manjiro's vantage point, relaying the scene to us and rooting it in Manjiro's experience.