In the book Gathering Blue, does Kira's society relate to the United States?

The short answer to your question is yes, Kira's society does relate to the United States.  

But to be more specific, Kira's society relates only in parts.  The society is not a mirror image of American society.  

Thematically, the story is about never giving up and working hard.  It has that "pull yourself up by your own boot straps" mentality.  Kira has to fight tooth and nail to prove that she can be a contributing member of her society, and she is surprisingly successful.  That theme is close to a standard American "rags to riches" thought process.  Work hard enough, and you can be successful.  

A more concrete reflection of U.S. society is the book's portrayal of committees and councils.  The governing body is exactly that -- a group of people.  It is not a single person like a dictator or king.  That relates to American society's legislative and judicial branches of government.