You are bringing the first quote a little out of context:
The idealists argue that the hexagonal rooms are a necessary from of absolute space or, at least, of our intuition of space. They reason that a triangular or pentagonal room is inconceivable. (The mystics claim that their ecstasy reveals to them a circular chamber containing a great circular book, whose spine is continuous and which follows the complete circle of the walls; but their testimony is suspect; their words, obscure. This cyclical book is God.)
So, the cyclical book is God in the context of "what the mystics claim".
The phrasing is a little ambiguous. It could imply that one of the mystics' claims is that there is a cyclical book, and another is that this cyclical book is God. Or, it might be the narrator's opinion that the cyclical book of which the mystics speak, is in fact -- in some manner -- God.
The separation between the claim of the circular room, and the definition of the book as God -- particularly, by saying "their testimony is suspect; their words, obscure," which obviously isn't the mystic's own claim -- makes it seem like a remark by the narrator. So the easiest way to read this is, the narrator is mentioning a belief of the mystics, and pointing out that in the context of their beliefs, the cyclical book is God. You could interpret that as, for example:
- Whether the mystics see it this way or not, their unshakable belief in a circular room and a circular book, effectively makes their belief in the existence of the book, their equivalent to belief in the existence of God.
- If such a book were indeed to exist, it would be a miracle, and could be seen as divinity itself.
- The mystics literally worship a circular book they believe exists as their God; however that is a minor aside, I only bring it up because I was telling you about shapes of rooms in the library, and a circular room is part of their myth.
The later quote is much more straightforward -- it's clear that we're talking about somebody being analogous to God, and not actually being God itself.
The statement here is that, somewhere in the infinity of the library, there must exist a perfect and accurate index. The Librarian who stumbles across it may be no different or better than any other Librarian -- but by having discovered the true index, he will be able to find anything in the Library he desires. This, in this story, is akin to Godhood.
This is a complex and unusual view of godhood -- one in which godhood is a thing of beauty and inspiration, the idea of finding order in chaos, and also almost impossible to touch or come near, because the library is so vast that the god, or the book, cannot be found.
There are parallels between the two usages -- first and foremost, both portray a God characterized by its ineffability, by the impossibility of finding it, and by the fundamental way the existence of such a God would change the world as the Librarians know it.
However, there isn't any particular indication of identity beyond this. Borges here creates a world and a philosophical thought-experiment by using concepts that are clear and concrete and intuitive; that's a lot of what makes the construction of the world so clear and compelling. There's no particular reason to look for a circular book and a man with a true Index, as being "actually the same thing," in some way.