As tempting as the comparison may be, let's not compare class structure to a legal code that legitimized the ownership of other human beings. That's really a bit of a smear and an insult to somewhat recent ancestors of people of African descent in America. Slavery is gone. Class divide is not and never will be gone. It's essential to the Capitalistic system, as out of control as it is now - it's not going anywhere.
People may find jobs that they like, but once they have children, they become sort of slaves.
They have to dump their children off elsewhere because they have to work at specific times.
Why don't they just quit?
They have to pay for the car they only need to get to work, they have to pay for the childcare they only need for work and the school they only needed to work.
People spend their lives making money for the big bosses. They get small fractions of what they make.
People don't think they're slaves. People have a little more slack...
People are either slaves, or slave-owners/ employees or business-owners.
Perhaps in a sense 'slavery' does still exist. However, assuming it did, it is a slavery within one's self. The overwhelming desire to obey social norm to continue in one, strict form of "reality", without a second glance at the world.
At least within America, people no longer honestly oppress each other, but it does appear that people are still slaves to the same group-think which they have victimized themselves to since the beginning of time. A rather vicious, endless cycle.