The Archive list syncs, but the actual books do not. You have to download them for each device you want to read them on. Only books purchased from Amazon will sync. You can turn synchronization off, so make sure it is on for devices you want to sync. Sync'ing, to Amazon, means keeping track of last page read, bookmarks, list in Archives, etc., not the content, per se.
You have to connect to Amazon to sync. Per the Amazon support page for Kindle for iOS:
"Delivering Library Content to a Device
The section of Manage Your Kindle titled "Your Orders" allows you to queue up items in your Kindle Library for delivery to your iPhone, iPod touch or another registered Kindle device. If the content is eligible for delivery to the Kindle application, your registered devices will appear in the "Deliver to . . ." pull-down menu.
Image of Your Orders section on the Manage Your Kindle page.
To send archived items to your iPad, iPhone or iPod touch from Amazon.com:
1. Visit the Manage your Kindle page.
2. Scroll to the "Your orders" section and locate the item you wish to download.
3. Click on the "Deliver to" pull-down menu and select your iPad, iPhone or iPod touch.
If your iPad, iPhone or iPod touch is connected to a network, your book will download and appear on the Kindle Home screen the next time you launch the Kindle application. You may be prompted to log into your Amazon.com account if you haven't already."
It doesn't help for backup, because each copy is encrypted for the device or app that it is downloaded for, and cannot be used on others. You could create a folder on the pc, and download for each device, keeping a separate folder for each device.