I do not know anyone that uses the terms “near” and “off”. This includes Pony Club A graduates, veterinarians and experience farriers. Just like people, horses have a left side and a right side. I may get confused but I also get confused as to which of my hands is left so I’m not really a good example. No one else gets confused because it’s pretty straight forward. Near and off, to me, is not - like near you…what if you are on the right side of the horse, is that now your near side? Or was that your near side before?
As far as reins, same thing - the left rein is in your left hand.
I have heard “outside” and “inside” used when directing which way to turn (ie - make your circle towards the inside of the arena) and for younger students that do not yet have a decent grasp on left and right (and possibly me…).
I can also straddle anything from either direction but that’s something I work on with riding also. My mare also can be led from either side because all of that is just tradition dating back to the knights.
As far as cars go, I do use driver and passenger side because people seem to get confused. I think it is more because cars aren’t living creatures so people do not consider them to have innate sides like people and animals.