The issue is the subject of multiple threads at Apple’s support forums, and has been verified by AppleInsider. Anecdotes hint that even some people within Apple may have suffered the problem, and that in some cases people are losing additional data such as Health statistics or Safari browsing data.
One possible cause of the problem may be connected to an original device that had been backed up while running iOS 9.0.1 or later, while the new device is running iOS 9.0. Other users have suggested perhaps iCloud is serving up corrupt backups.
Regardless, one user-proposed solution is making sure both the new and original phones are updated to iOS 9.1, running a fresh backup of the original device, and then resetting the second phone to factory settings. Only at this point is it theoretically safe to restore from iCloud on the new hardware.
Another solution is to use local iTunes backups, especially in cases where the original device will be missing or non-functional when it comes time to restore to the new device.
Apple Support tells customers that the company is aware of the problem, and is working to find a solution. No timeline for such a solution has been officially announced.