Will the decorations I get for my Archive Bookshelf work on the Pet Bookshelf, and vice versa?

Yes! Any Bookshelf Customisation item can be applied to any Bookshelf, whether that’s the Pet Bookshelf or the Archive Bookshelf. You can’t use the same item on more than one Bookshelf at a time (which means if there’s a plushie you reeeally want on all your pets' bookshelves, you’ll have to buy more than one), but all items bought for your Bookshelves will be part of the same inventory.