Andersen's use of Point of View is all over the freakin' place.
You want third person? Check out any of his fairy tale-ish stories, such as "The Traveling Companion," which tends toward third person (limited omniscient) with snippets such as, "Johannes was not worried or frightened about what was going to happen." (7.61) Well, you get on with your bad self, Johannes, and enjoy that chillaxed way of life. We wish we could be so worry-free. Our never-unbitten nails would appreciate it.
What's that, you wanted third person omniscient? Hang on, he's got that too. Check out "The Magic Galoshes," in which he transports us into the heads of the various characters who put on the magic galoshes… and are treated to whatever they wish.
Hungry for some first person? Read "The Pact of Friendship," in which our central first person narrator tells us about his life growing up in rural Greece and how he came to have an adopted sister and a best buddy. For a peripheral first person narrator, see "From the Ramparts of the Citadel," where "we are standing on the ramparts of the citadel" (40.1) opens the story, but then we actually hear about a prisoner nearby for the rest of the tale.
Oh, were you in the mood for second person instead? Even this elusive perspective can be found in Andersen's tales. "Auntie Toothache," the last tale in the collection, starts out with the narrator writing, "Where did we get this story from? Would you like to know?" (156.1-2) And references to "you" are then peppered throughout the rest of the tale. Whoa, there, we're starting to get dizzy from all these perspective shifts. Hold up, we're going to go put our heads down and take some deep breaths.
Okay, back. Now let's tackle Andersen's craziest narration strategy of all. In some tales, Andersen tucks a few first person and second person references into the middle of what's otherwise a standard third person story. Like, in "On the Last Day," the bulk of the story is written in third person, as we hear about this one soul's issues with dying and trying to get into heaven. But then, Andersen writes, "Every one of us will on the last day and hour of life here on earth draw back in fear and humility from the glory and splendor of heaven" (59.33). Okay, Andersen, if you say so.
And then there's "The Philosopher's Stone," which is mostly told in third person, but it begins with this exchange: "You remember the story about Holger the Dane? I don't want to tell it to you; I am only asking to find out if you remember how Holger won great India" (80.1). So, in this tale, we get a double whammy of first and second person intruding into an otherwise third-person-narrated tale.
When it comes to narrative technique, Andersen's tales are practically a buffet. Have a little of this, a little of that, and come back for more if you want. Delish.