Now it may be a regional thing, but I’m pretty sure anyone mentioning ‘skid marks’ hereabouts is talking about stainage of the undercrackers. Normally occasioned when one had to use a Public Convenience with the get in / get out / avoid the glory hole mentality . Added together with that old style shiny toilet paper.
I can sort of see where they are coming from with the name as in marks left by skidding in the road , but to me it just means soiled pants, neither of them I would have thought you’d want associated with a driving school.
As an aside , in our family they have come to be known as ‘the workings of the ship’ in honour of a line in one of Patrick O’Brien’s Aubrey / Maturin novels