For a book to appear in a bookshop or library, it has to first be bought from the wholesaler. The wholesalers will not buy a book from Amazon or any other online retailer because they are not offered it at trade discount (35-60%). This discount is the leeway the sellers need in order to make a profit on selling the book and still leave enough for the author royalty.
The ISBN on the book tells the wholesaler who the publisher is and what format the book is produced. They put this information in their catalogues so bookshops and libraries can find out about the book before they order it. The wholesaler receives the order and passes it on to the publisher who supplies the book at the discount price.
It is essential for this process that a full 13-digit Neilson ISBN is applied to the book. A publisher who uses a shorter number (such as the free online ones) is only making the title available online in their own marketplace.
If you want to see your book on the shelves in shops and libraries, it must bear a full ISBN.
Waterstones is different again. They require 60% trade discount which is too much for small hybrid publishers in the main. So, to get round this, a customer can order a book from Waterstones who will order it as a one-off or special order. The customer may collect the book from Waterstones but will be required to pay the p&p in addition to the cover price. The same criteria regarding the ISBN apply, as Waterstones will only order from the wholesaler’s catalogue.