I don't know much about ISBNs. I know they are an identification number, and that each one must be unique for a given version of a book. Also, if I understand correctly, whoever bought the original ISBN is listed as the publisher of the book version that uses that ISBN. But that doesn't give them any copyright license, right?
An ISBN refers only to the specific edition of a book that it's attached to. It has nothing to do with copyright, which is a completely separate issue.
Can someone who knows more tell me what the advantages are for owning the ISBN as a publisher?
An ISBN is essentially a tag which allows everyone involved in the selling of that book to ensure they're working with the right book. It's used in catalogues, online and off; it's used to produce sales statistics and royalty statements, it's used for ordering and stock reports and production documents. All sorts of stuff. It makes sure that different editions of the same book aren't confused, and that books with identical titles from different authors are separated too.
This is really confusing me since it is split from copyright in such an odd way. What rights does owning the ISBN confer?
The ISBN has nothing to do with copyright. Owning an ISBN doesn't confer any rights at all in this respect.