Your second image shows that 'This disk is used for Time Machine backups'. That suggests you have somehow used it for Time Machine. Is this the case?
At the root of this disk there will be a folder called "Backups.backupdb" if it is/ was used for Time Machine. Let us know if you have that.
I think that is the reason for not being able to install. You may need to remove that folder, however it is difficult to do because the Backups folder has many complex permissions controls.
You could try renaming the disk in Finder, but I doubt the name is the reason for it not being 'installable'. Do that by opening the 'computer' view in the Finders 'Go' menu, select the disk, tap enter to make it editable & rename it.
Another possibility is to create a USB bootable installer on another disk, then boot from that & see if the internal disk can be selected as an install destination…
Either of these will create a bootable installer…
http://diskmakerx.com/
https://support.apple.com/en-gb/HT201372
Reboot holding alt, select the USB install disk & see if that can install to the internal.
If you have spare externals disks I would make a full backup first, it seems like you have a weird setup that may be safest to erase & start over with.