What do Isabel and America have in common in the book Chains?

Your question is about the similarities between Isabel, the main character in Chains, and America, the country.
Isabel and America have several similarities, including youth, betrayal, the desire for freedom, and the struggle to achieve it.
When Chains opens, the United States of America doesn't exist yet, as such. The story plays out against the backdrop of the Revolutionary War. In the beginning of Isabel's personal story, she's only 13.
The betrayal of Isabel by the people around her mirrors the struggles of Revolutionary America. Again and again, she's cast down and kept from getting what she's promised. Her original owner's sibling refuses to free her. Her new owners abuse her. The Colonel she spies for doesn't help her when her sister is sold away from her.
In comparison, a major issue in Revolutionary America was the restriction of the freedom of movement. The Proclamation of 1763 kept colonists from moving to new territory and was very unpopular. In the same way, Isabel was kept from going where she chose. Tax laws put a burden on the colonists, the same way her abusive master's demands were a weight on Isabel. There was physical violence in both cases, too.
The desire for freedom is illustrated by Isabel's thoughts throughout the novel, her decision to pursue reading, and her willingness to turn to the British side of the war if they'll free her—even though that is prevented and she continues spying for the Americans.
Isabel and America both have to struggle to get what they deserve. Isabel spies for the Americans, fights her abusive owners, and steals a pass that declares her freedom. America's fight is illustrated in the battles of the Revolution, which are entwined with her own.
Anderson uses primary source documents like Common Sense and letters from the founding fathers both to set the scene and illustrate the parallels between Isabel and the USA. There are excerpts at the beginning of each chapter.
http://courses.missouristate.edu/ftmiller/proclamationof1763.htm

https://www.historycentral.com/Revolt/causes.html

Readers can see several parallels that exist between Isabel and America. 
First, Isabel is a slave. She has masters that are ruling over her. When the book begins, America is also being ruled over by a master. That master is Britain and its king. The book begins in May of 1776, and the United States didn't declare independence until July of 1776.
Second, Isabel is trying to gain her freedom throughout the book. She attempts to get her freedom in a variety of ways, and she succeeds by taking matters into her own hands. America did the same thing. The colonies attempted to gain independence through all kinds of legal channels, but only succeeded in gaining independence by taking the initiative and forcing it to happen.
Third, Isabel is a young character. America is a newly formed (young) country.  
Fourth, Isabel is abused by her master. America is abused by Britain. For example, "taxation without representation" was a major sticking point for the founding fathers.
Fifth, Isabel receives help from Curzon in her efforts to attain freedom. America eventually gets help from the French in order to secure independence.