I'm still not sure how you're getting that display (perhaps you've renamed the "Data" folder to "... Family Time Capsule," but in any event, it does show both sparse bundles properly.
And you can open your sparse bundle.
The spinning icon under DEVICES in the sidebar means Time Machine is doing a backup. That's a problem: Migration Assistant automatically uses the most recent completed backup.
So, turn Time Machine OFF and cancel the backup.
If the backup has completed, you may have to re-mount the disk image (double-click the sparse bundle). The new backup will be date-stamped after the 9:23 one you want (per the second sample), so delete the new one. If the backup didn't complete, it will still have ".inProgress" in the name and Migration Assistant will ignore it, so you don't have to delete it.
Try Migration Assistant one more time. If it doesn't find your backups, they may be damaged. See if you can repair them, per #A5 in Time Machine - Troubleshooting.