And very often where the insurance does cover 'any vehicle' the policy specifies 'in an emergency only'.
I contacted my insurance company and they gave the example of the driver being taken ill at the wheel, I as a passenger would be able to drive him home, to the gp or to the hospital.
legend007 seems correct from my experience i.e you are covered to drive but only 3rd party BUT there is almost always an exclusion which says that this only applies where the car is borrowed but not hired. If you rent a hire car/ van etc you have to be separately insured. You are also (of course) still only able to drive vehicles of a type covered by your licence - so manual/ automatic considerations still apply also.
I might also add that it's not only comprehensive policies(there's no such thing as fully comp but people still call it that) that allow you to drive other cars on a 3rd party basis.
Most companies, certainly ones underwritten by UKI, will allow you to do it even if your policy is 3rd party only. If you've already been assessed for 3rd paryt risks in your car why should it be any different in anyone elses?
Ethel is correct though, some companies will only permit it in an emergency provinding the car isn't private hire/rental.