Add Home Folder Hard Drive to Favorites and Desktop
I've solved it myself. I just tried to drag Storage into Favorites now and it works for some reason!

I'm thinking that it's because I (a few days ago) wrote found a command that was meant to remove nobrowse. (I don't recall exactly where I found it or exactly what it was, so I can't paste it here. Sorry).

After running the command, Storage suddenly appeared on my Desktop, even though (as it can also be see in the question) OS X didn't mention anything about the drive having the flag nobrowse. Either way, after Storage appeared on the Desktop, I still couldn't make it show up under Favorites. Until now. I hadn't tried to reboot earlier, so that might be why it suddenly works.

Thanks to everyone trying to help in the comments above.