I don't think there is such a thing as a special exception to copyright law given to one website.
There is fair use.
I think what happened with archive.org is that it became popular, and also popular to think of it as fair use. It's a social phenomenon of acceptance that does not have any legal bearing.
Companies that don't want their stuff 'archived' can and do take action to enforce laws about digital libraries. For example the book that helped teach me programming was Turbo Pascal DiskTutor. You cannot simply download that one on archive.org. You have to get on a waiting list and 'borrow' it when it's available.
The fact that they made it so there is apparently exactly one digital copy available for 'borrowing' makes me feel that the digital library laws are invalid. It should not be legal to enforce only one copy total when it is possible to make it ten just as easily.
Anyway, there are lots of sites like YouTube that would not exist without encouraging users to violate copyright. This was the whole reason YouTube got big in the first place. It was only after they had a massive library of content and users that they started really playing ball with distribution companies.