A library is publicly accessible, just like the Internet Archive. What's the difference?

actually quite a few:

https://www.bl.uk/help/what-is-library-privilege

check out the section “ Your obligations when using the Library Privilege Service”

Isn't IA officially registered as a library? Why shouldn't they have the same privilege?

from what i know on this subject only “national libraries” have a special carve-out for copyrighted works. and even then the rules are very strict.

National to whom? I believe IA has offices in at least three jurisdictions.

> A national library is a library established by a government as a country's preeminent repository of information. Unlike public libraries, these rarely allow citizens to borrow books. Often, they include numerous rare, valuable, or significant works. A national library is that library which has the duty of collecting and preserving the literature of the nation within and outside the country. Thus, national libraries are those libraries whose community is the nation at large. Examples include the British Library in London, and the Bibliothèque nationale de France in Paris.

https://en.wikipedia.org/wiki/National_library