What is special about these screwdrivers that makes them ESD safe compared to other screwdrivers?
The data sheet for that T4741SESD screwdriver set explains that:
The handles are [...] electrostatically dissipative to DIN EN 61340 5,1 & 5,2
That is the difference.
Therefore if you working at an ESD workstation, wearing an earthed wriststrap or footstraps etc. and pick-up one of those screwdrivers, your body will provide a high‑resistance path to ground, to discharge any accumulated electrostatic charge on the handle (or shaft) of the screwdriver. The same applies if the tool rests on a grounded ESD mat.
The idea is to remove any electrostatic charge from the tool and its handle, which could possibly otherwise cause ESD damage if it was moved close to ESD sensitive devices. Most ESD policies prohibit "non-process essential" insulators from ESD workstations, and ESD policies (and audits) may classify normal insulating tool handles as an example of that.
Is it possible I have inadvertently destroyed the ESD safe properties of the screwdrivers by removing the brown coating on the blade/shaft, or could this just be some oil to protect them from rusting?
No, you haven't damaged the ESD properties because the shaft does not have special ESD properties. I agree that you have likely just removed some kind of oil coating, since it came off with only light rubbing. The same thing happened with some ESD‑safe hand tools which I received from Farnell - they came with a coating of light machine oil on the metal parts.
Note: The static-dissipative handles on ESD tools means that they should not be used where they could be in contact with high voltages, since there is a deliberate electrical path (albeit a high resistance one) through them, to the person holding them.