The id a book is bound to is the id you are logged in as when you redeem the book.
The library itself doesn't require an adobe id.
If you checked it out with OverDrive Media Console, OMC requires an Adobe id to redeem the books, too

but you should've been prompted for one. The newer versions I believe use some complicated alias with their servers instead... but that is still a service of the app, not the library.