Based on the Fugitive Slave Act of 1850, all American citizens were compelled to assist in the recapture of runaway slaves. During the 19th century (the setting for Lyddie), many Southern slaves made their way to Canada and the Northeast through the Underground Railroad. Essentially, the Underground Railroad was a network of secure routes and safe houses African American slaves utilized in their journey toward freedom in the North.
Those who manned the safe stations were called "conductors." Most conductors were free blacks, but some were white abolitionists. Quakers were some of the most fervent supporters of the Underground Railroad in the 19th century. Although many Quakers owned slaves in the 16th and 17th centuries, the tide began to change during the 18th century. Then, many Quakers began to speak out against the scourge of slavery, and many others participated in the abolition movement.
In chapter 6 of Lyddie, we learn that the Stevens family originally housed Ezekial Abernathy, a fleeing slave, at their own farm. However, when it became clear that the Quaker farm was being watched, Luke Stevens moved Ezekial to Lyddie's cabin. Basically, the Stevens family were conductors on the Underground Railroad. Their attitude toward slavery was indeed revolutionary for their time and definitely ran afoul of slavery laws; if caught, the Stevens family would have been imprisoned and/or fined.
Meanwhile, Lyddie is similar to a slave in the sense that she has little power over her own life. By working for the mill, she agrees to submit to all the rules and regulations imposed upon her. As a lowly mill worker, Lyddie has few rights. Powerful corporations decide how the mills are run. To increase profits, these companies can prolong working hours and lower base salaries with impunity. This is the main reason Lyddie's peers sign a petition that advocates for better working conditions for all mill workers. Lyddie is similar to a slave in that she has little personal autonomy and few options to better her life.
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.