The Quaker question is a bit confusing. It's asking about an attitude being different than a law. I'm not sure I see the connection. The Quakers during Lyddie's time are not that much different than the Quakers in modern day times. The reason for that is "Lyddie" is taking place during the industrial revolution which marks a beginning of sorts to the modern age. New technology, business, and manufacturing techniques were being developed quite rapidly. Cities were becoming huge. The Quaker attitude was, and is, to stay apart from that. The tools that they had been using for generations still worked, so there is no need to change. They saw all of that new development as getting in the way of living in close harmony with nature, family, and faith.
Lyddie is similar to a slave because while working at the tavern, she has been forced into a job she does not want in order to pay off debts that are not hers. She longs for freedom like a slave. A slightly different freedom, but still freedom. In Lyddie's case she wants to be financially independent in order to make her own choices. She doesn't want to be beholden to anybody. Also similar to a slave is her high work ethic and desire to be something more. She loves learning and sees that she can gain further independence with more knowledge.