Like the rest of the books in the Harry Potter series, Fantastic Beasts fits pretty neatly into the Fantasy genre—when Newt's talking about what to do when you've been stung by a Billywig you know you've gone into the realm of strange and other-worldly settings.
The time to sit back and suspend a little disbelief is now.
We'd also like to note that though the book isn't strictly a fairytale or a myth, it does draw pretty heavily from the world of (Muggle) folklore, legend, and mythology. Be on the look out for lots of well-known creatures Muggles from stories such as centaurs, pixies, leprechauns, and unicorns. Of course, Muggles know all too well that this beasts are strictly fantasy. Just like this book.