This also fails to take into account that ISBNs also contain the publisher ID in them. So identical copies of a book could have different ISBNs depending on which markets they are sold in.
They don't contain the publisher name, but ISBNs are usually purchased in blocks of 10 or 100 or 1000 or whatever by a single entity, which is often a single publisher or corporation.
However, within the block publishers can assign ISBNs to different imprints.
I'm not sure this is the case, I got my ISBN range through my government national library service, I could be wrong but when you let them know what the book is you are publishing they ask for the Publisher name, though I am guessing as the service is free and it only applies to New Zealand books and publications.