This is a silly question but what is the "official" term for what is carrying this hard drive?
I need to know if I can purchase it separately from the hard drive.
This is known as a hot-plug "drive carrier" or sometimes "drive sled", "tray" or "caddy"...
And yes, they can be purchased separately from the drive. There are companies that specialize in providing aftermarket or replacement drive caddies. Ebay is also a popular source.
What type of server or enclosure is that carrier from?