여기에 책이 네권이 있어요 this sentence has two subject; 책이 & 네권이.
The expression "여기 책이 네권이 있어요." is perfectly fine and idiomatic.
No, subject is still book.
-이 has several different usage.
Most commonly used it obviously to indicate which is subject.
Another usage is when you try to emphasize.
One example can be:
"도대체 우리 행동이 무엇이 잘못되었다는 거야?" Quoted from Naver Dictionary.
Hachiko's expression is emphasizing there are indeed four books.
Whether Hachiko had such intention to mean it, there is not enough information provided in the original post.
So it is very critical to provide as much information as possible for other members to help correctly.