Ive heard this repeated on the BBC before, but it isn't true, at least not for my driving lessons 2 decades ago. I just got told everytime to look over my shoulder for cyclists before opening the door. But never have I heard of anyone being taught to specifically open their door with their right hand

Having taken my driving lessons and test more recently, a couple years ago: I've been told by my instructor that not doing the dutch reach is one of the few items that can result in an instant fail. I think opening the door that way only got formalized somewhat recently.

To be fair, the BBC is institutionally anti-cyclist, so they may have mis-represented the "Dutch Reach".

I can't see why it's not taught and used everywhere as it encourages and facilitates the checking behind you when opening a car door. Rather than focussing on "left" or "right" hand, I find it more useful to just always use the furthest hand from the door so the same idea applies if you're driving or a passenger.