Yamibo - Darkness, the Hat, and the Travelers of the Books (2003)

ヤミと帽子と本の旅人

aka Yami To Boushi To Hon No Tabibito · 暗与帽子与书之旅人 · Yami to Boushi to Hon no Tabibito · 어둠과 모자와 책의 여행자 · Ями, Шляпа и те, кто путешествуют по книге

Hatsuki is a highschool student living with her sister, Hatsumi, who she has a huge crush on. On Hatsumi's 16th birthday, she is suddenly surrounded by a green light and disappears in front of Hatsuki! She manages to follow Hatsumi with the help of a being resembling a fat baby chick (literally), ending up in a place called "The Great Library", which is full of different worlds stored in books. Hatsumi wasn't there, though, so the search for Hatsuki's great love begins and involves traveling from book to book.