Cannot enter Time Machine on rescued drive from dead Time Capsule
As noted in some comments, I would not rely completely on that disk since it was part of a failed system. 

However, since it seems like the time machine files are stored inside a disk image, it should be possible to use Disk Utility to restore that image to a new empty harddisk, as long as it is at least the same size (like 2 TB in your case). When that is done, I would expect the new disk to be recognizable for Time Machine as as a backup disk, and continue your backups from there.

First, connect a new, blank hard disk. Open Disk Utility and make sure the new disk appears in the Disk Utility Window. Now drag the icon of the disk image from your old backup drive that contains the Time Machine partition (named 'mac' in your first screenshot) into the left sidebar of the Disk Utility Window. Then you are able to restore that disk image onto the new blank disk. The new disk will now contain the complete filesystem of the disk image and should be recognizable for Time Machine.

Be aware though, if the file system is corrupted on your old disk, it would mean that continuing backing up on top of that file system would continue to be corrupted, and therefore you are at risk of having an unusable or incomplete backup, even after several incremental backups.