The riddle
What Has a Spine but No Bones?
The back edge of a book
A book's spine is the narrow bound edge that encloses the inner edges of the pages and faces outward when the book stands on a shelf. Wiktionary lists that as one sense of spine, next to the backbone of an animal, and adds that it typically carries the title and the names of the author and publisher. The riddle removes the animal sense with the words "no bones" and leaves the book sense as the only fit.
When spines were hidden
Wikipedia's article on book design says early books had no titles on their spines. They were shelved flat with the spines turned inward and the titles written in ink along the fore edges, the side opposite the spine. Modern books show the title on the spine instead. In the United States, the United Kingdom, Scandinavia and the Netherlands the spine text runs from top to bottom, while many continental European and Latin American countries print it from bottom to top.
From a thorn to a book
Wiktionary traces spine, through Old French, to the Latin spina, meaning a thorn, a prickle, a spine or the backbone. So a pointed thorn and a backbone were already one word, and later the same word came to name the edge of a bound book.
Similar body words
Objects borrow body words all the time, as in what has teeth but cannot bite. Asking a group to list more of them, one object at a time, makes a quick game.