Re: It beggars belief...
"As I understand from the very broad definition, If you're getting paid, it's an enterprise. If you're not getting paid, it's a car share."
I don't think it unreasonable to expect some remuneration towards petrol & wear and tear (as defined by your government's own tax office).
Car sharing without is fine if you drive half the time, and I drive the other half, but if you always do the driving then I'm not contributing appropriately to the arrangement - paying 50% of the relevant milage rate seems fair to me - and probably good for you as well.
Carry two people to work (which I could easily have done at a previous job and you can either cut the costs in 3, or maybe everyone you carry is happy to put in 50% - in which case your own commute is free.