Tme machine is not dependent on the user.. it is local machine with specific access to backup the whole drive including all users.. but even the early TC will block two users access to the disk at the same time on the same computer.. I think you can access the TC disk from two different computers. I do not know exactly why the limitation was set but it is there for sure..
The 6th Gen AE does change the scenario for TM being able to backup to external drive plugged in.
My understanding was that the 6th-gen Airport Extreme could take the place of a Time Capsule. Is this not the case? How is the way an Airport Extreme treats a network drive different from the way the TIme Capsule did? Is there just some setting somewhere I've missed?
I have seen nothing to suggest the AE should behave any different to the TC.. but it will depend on the security access you have set to the drive.. what is the AE set to?? (what was your old TC set to??)
I don't have a Gen6 so I am not sure what is possible.. but normally security is set at 3 or 4 levels.. guest access read and write.. most open.. least security.. to user accounts where access is limited to specific users with account details on the TC.. this is the most secure.. with AE password and disk password as in between levels.
Thanks for the response. It sounds like the behavior I'm experiencig (especially accessing a network share from two accounts on the same machine) is by design. For the time being, I've moved back to a local TM backup and will just use the Airport USB port for a basic network share that won't require multiple user access at the same time. I may return to this topic at a later date, but as of right now, your answer solves the problem for me.