Time Capsule stuck on makingbackup disk available

I experience issues like this from time to time. Although running a later version of MacOS (10.13), most of the steps should be similar.

What I normally do is:

• Use AirPort Utility to check if the firmware on the AirPort device is current, then restart (even if the firmware is up-to-date.
• Run Disk Utility on the Time Capsule backup volume (whether external or integrated in Time Machine)
• Restart the main computer, holding down the CMD + R keys to start up in recovery mode, then run Disk Utility on all storage volumes that are backed up with Time Machine.
• Disconnect the Time Capsule volume, then reconnect. Hold down the Option key, then under Time Machine in the menu bar, run Verify Backups.

Situations like this can be frustrating, and it may take some time to do this, but I find it usually helps. For some other tips, Apple has some suggestions here.

A1: I do believe TM uses UUIDs to identify backup servers. There shouldn't be any issues with duplicate names.

A2: Renaming the disks would probably help you identify it, but it is unlikely that it would solve this problem. As mentioned in A1, TM uses UUIDs to identify disks.

A3: I would try removing the disk through the "Select Disk" button below the disk list (this does not delete your existing backups) and reselecting it.

I would believe that this was likely caused by a graphical glitch.

Good luck.