If there's no slavery in a fictional setting, then you're not diminishing what your character overcame. You're simply not including that one thing. There's still many other obstacles that can be overcome and many other ways your character can be an underdog. Nasty rivals or superiors, bullies, abusive parental figures, being orphaned or or an outcast, being born to a poor family or low part of society, being framed for a crime, having accidentally committed a crime, being part of a rebellion, and so on. These are just what came to be with about thirty seconds of thought.