What did Isabel and America have in common?

Like Isabel, America is just starting out in the time period when the book begins—indeed, it is not even independent from Great Britain, and even by the end of the book it is only recently acknowledged as a country. As such, Isabel feels some affiliation with America as an entity in its early stages.
As a slave, Isabel is extremely unhappy with the way she is treated by her owners. This reflects the dissatisfaction felt by America about the way it is treated by its British overlords—the young country is suffering under issues of being taxed without representation, essentially being asked to contribute without being recompensed. As a slave, Isabel is confronted with this same issue.
It is no surprise, then, that Isabel wishes to break free of her owners. This is echoed by the young country of America seeking freedom from its mother country, knowing that its worth is greater than what the mother country acknowledges. However, neither Isabel nor the USA is able to achieve its goal without resorting to some level of violence. Isabel has to escape her house by force, faking her valid documents and fleeing from the state. Meanwhile, the US achieves its independence through an act of war.

Both Isabel and America are relatively young.  Isabel is a young slave girl, and America hasn't even declared independence when the book begins, which makes America a very young country by the time the book ends.  
Both America and Isabel are also unhappy with their respective masters.  America is very unhappy with British rule and the treatment from the king.  Things like taxation without representation are big issues for the fledgling country.  Isabel is equally unhappy with her master, Madam Lockton.  Madam Lockton is incredibly abusive to Isabel and Ruth.  She beats both girls at various points in the novel, and she even has Isabel's face branded.  
Because of their harsh treatment at the hands of unfair rulers, both America and Isabel deeply desire freedom.  Additionally, in order to achieve that freedom, both Isabel and America are forced to take that freedom by force.  America declares independence and fights a war to keep it.  Isabel breaks herself out of Madam Lockton's house, fakes a document that states that she is free, breaks Curzon out of prison, and escapes New York.