Can Monkeys own copyrights?

Seems that a monkey was using an idle camera to take self portraits:

So, when websites started using the images, you can imagine what happened.  The Carter News Agency is asking sites to remove the images.  The justification: TechDirt doesn't own the copyright. 

Question is: who does.  If no one, then what authority does Carter have in asking TechDirt to remove the images.  Does the request imply that Carter think that it owns the copyrights?  If so, how.  If not, then the request is baseless.  In fact, making false DMCA take down claims (which has not (yet) happened) is itself illegal.