“Not every white Southerner was a slaveholder, and not every white Northerner opposed racial discrimination”

This means you should avoid treating geographical categories as if they represented uniform populations.

For example:

South

A white Southerner could be:

a wealthy plantation owner;

a small farmer who owned no slaves;

a poor laborer;

a merchant;

a person who opposed slavery;

a person who supported slavery.

Therefore:

Southern is not the slaveholder

North

Likewise, a white Northerner could oppose slavery but still believe that Black people were socially inferior.

So:

opposition to slavery is not the belief in racial equality.

This distinction is particularly relevant to the argument about the persistence of social attitudes after emancipation.