To an extent. A wizard would consider himself distinct from a sorcerer who would consider herself quite different from a warlock or cleric. They would each probably refer to themselves in specific terms to distinguish themselves as such. A wizard might refer to himself as a Conjurer (though not of cheap tricks) but would identify as a practitioner of wizardry, which is distinct from sorcery.
The same cannot be said of the average person living in the world, who would probably use the terms almost interchangeably. In the common vernacular, they're effectively the same thing.
That said, a fighter almost certainly would not identify as a fighter. They'd probably call themselves a swordsman, mercenary, knight, or somesuch. Others would do the same.
Class is not a known quantity in my campaign settings per se, but the abilities and trappings thereof can be, and oftentimes those are identified using a similar nomenclature.