Do I need to ground Ethernet cable run beneath a pier & beam house?
The cable you bought looks fine.  You don't need STP (shielded twisted pair) wire.  From what I've heard, it's mostly used in industrial environments with a lot of electrical interference issues.  Won't really apply to residential environments.
I don't think that's a drain wire since the cable is not shielded. That extra wire is not copper in the picture, so I doubt it's a ground.  My guess is that it is there to provide pull strength when fishing the cable through conduit.  Cat6 is stiffer and less flexible than Cat5, so it will have more stress when pulling.