Are authors identified by the chapters they contributed to in multi-authors books I wrote 4 chapters of a book, and three more authors wrote other chapters.
I was expecting that each chapter would reference the author who wrote it, but the editor says it is customary not to do this, so the book will have us four authors as collective authors of the book, without any reference to each chapter.

Is this really customary?
It seems so to me. YMMV by the field, but I can't recall reading any books cited like that, except for compilations of essays where the "chapters" are really papers that are functionally distinct from each other.

Presumably you read a lot of works in your field. Go grab 10 books with multiple authors and see if any of them have a capter-by-chapter authorship breakdown.