They've obviously gone to great lengths to protect his identity.
My parents recently spent an evening with Jeff Goldblum and he said The Stig never took his helmet off and so he never saw who he was.
You really think he gives instructions from underneath a helmet 100% of the time or more likely the 'celebs' agree not to name him?
Although it's likely the stig is never the same person and they get an instructor in for the day that isn't always the same or collins.