What is this lamp base wire clip and how do I remove the wire without breaking the clip?
That is a Heyco bushing. You need to squeeze, preferably with a needle-nose pliers, in what I would call a top-to-bottom position considering the orientation of how it is installed.

If you squeeze just right it will come out no problem and will not damage the cord.

Here is a decent video explanation: 
http://youtu.be/G1IWQB3mQrw

Here is the manufacturer's guide:
http://www.heyco.com/img/SR_Assembly_Guide.pdf

Rest assured, it CAN be done with a regular pliers if you're careful. I've done it a million times. 

Also, keep in mind, to replace a socket you really should not have to remove that bushing if the cord is in good shape.