First Person (Peripheral Narrator)
This one's a tiny bit tricky, since there are a couple of different voices at play in this book.
The main one is the author's. That's Newt Scamander—he's the one telling us all about his life, his travels, and these seriously fantastic beasts.
But really, he's mostly telling someone else's story—the story of the beasts themselves and their history in the magical world. That's why he's a peripheral narrator. He's not giving us an account of what's he's doing. He's telling us about other people (or, in this case, beasts).
Then, of course, there's the foreword by Dumbledore (which is also in First Person) and which addresses the reader directly. His goal is to explain the benefits of buying this book and to tell you why you to support Comic Relief U.K. (Hey, it worked. We bought our copy.)
And last, but not least, is the voice of Harry and Ron (and even Hermione). Their notes in the margins add yet another First Person voice to the mix. They're speaking for themselves and often contradicting what Mr. Scamander is trying to get across.
Their scribbles also serve as little in-jokes with the audience because we've (presumably) read about their other adventures and get exactly what they're talking about.