I think this is definitely a thing. But I also think for a lot of journeys the choice of transport mode is made out of habit. Some people just need a nudge to consider alternatives.
Maybe low income families who are in receipt of council tax reduction and / or universal credit should be exempt from any surcharge on additional permits?
That article looks misleading to me. It makes is sound like charging more for 2nd and 3rd permits is a new thing, but looking at the current charges it looks like they introduced this last year. Also, note the exemptions: