Typically, when a class is applied to a Fate Core character, it is their high concept (sometimes with the race/species tacked on to it). I’m guilty of this as well, as can be seen in my Star Wars Fate Edition hack (although I suggested adding an adjective to help further differentiate). The problem with this is the same problem I have with aspects in general: they’re usually either too broad, or not broad enough.
Here’s an Option for Class and Race/Species Mechanics
Take what Fate Accelerated Edition (FAE) uses as a description of aspects: they describe what you can do. Approaches (skills), on the other hand, describe how you do it. Meanwhile, Stunts are what makes you unique and awesome. I think the description of the aspect is better defined in FAE, but we’re talking about classes and races here.
Races typically describe “what you are”. Well, that’s not really covered by either approaches or aspects, but stunts say what make you awesome and unique, and sounds like a fabulous spot for a racial description. Since Fate has 3 stunts to start with, a racial stunt could be designated as one of those stunt slots, with the other two considered free.
But classes are awesome, too, though maybe not always unique. One of the two available free stunts could be used for the primary class benefit. Sorcerers need magic to cast, warriors need to be tough or powerful (or both) and rogues need to be stealthy. But a warrior can be stealthy and use magic, and sorcerers can often swing around a sword and hide under a table. The stunt is what makes them awesome at it, what sets them apart from others who just have a high level in that approach/skill.
This doesn’t remove the racial/class inclusion in the high concept, but it means that the high concept is free to be something more unique than just a listing of traits, as well.
In truth, there are many ways you can represent classes and races in Fate Core. This is just one.