Hard Drive I replaced still showing up in Disk Utility and as a Startup Disk
It's actually not a phantom drive; there's a 128GB SSD that originally came with the Mac mini to be used in conjunction with the HDD as a "Fusion Drive."

From Everymac.com:


  This model also could be custom configured with a 3 TB hard drive for
  an extra US$150, a 1 TB or 3 TB "fusion drive" -- which combines a
  traditional hard drive with 128 GB of flash RAM -- for an extra US$250
  or US$400, respectively, or a 768 GB SSD for an extra US$1300.
  (emphasis mine)


You have a couple of options...


You could just format the drive to get rid of the boot partition and not use the drive - diskutil eraseDisk %noformat% disk0
You could format it FAT32 or HFS+ to make a separate drive for 128GB of storage to be used however you like - diskutil eraseDisk fat32 Untitled GPT /dev/disk0
You could re-do your installation and create another "fusion drive" that would be 1.13TB in size.  You will want to backup all your data, create the Fusion drive and then restore to the new combined drive. (I personally went this route)