This is because the (session) storage is associated with the document, not the window. Or more specifically, it is initialised depending on the origin of the current document:
https://www.w3.org/TR/webstorage/#the-sessionstorage-attribute
When a new Document
is created in a browsing context which has a
top-level browsing context, the user agent must check to see if that
top-level browsing context has a session storage area for that
document's origin. If it does, then that is the Document
's assigned
session storage area. If it does not, a new storage area for that
document's origin must be created, and then that is the Document
's
assigned session storage area. A Document
's assigned storage area
does not change during the lifetime of a Document
.
Of course, that every window has its document and every document has its window1 is implied only and stated nowhere in the WebStorage spec, nor is accessing the relation made explicit.
So when they say "Document's Window object", they refer to the document's browsing context, accessible as document.defaultView
, which are defined in the HTML spec (right next to window.document
).
1: which isn't even completely true, they might have associated documents/windows, but in all contexts where we access a storage they will.