You are correct. There is no knowledge roll that will tell you what the guy walking down the street in front of you has as his class.
Racial abilities should be under Knowledge Local.
Class abilities should as well, but only if you have something to tell you what class that PC/NPC is, as many of them have no outward sign.
A guy in leathers with a rapier could be a bard, a rogue, a swashbuckler, a fighter, hell... even a barbarian.
Until you have something to guess at what "class" an individual is, there shouldn't even be a roll to get anything other than visuals.
For example, even once they enter a rage, you don't even know for sure what they are. They could be a ranger, a skald, a barbarian, a cleric, an inquisitor... so that level of knowledge check is probably not settled by a dice roll at all.
That said, religion to get an idea of what divine casters are capable of, arcana for arcane, etc. is reasonable.