When I take a shower the dishwasher fills with water and so does the kitchen sink, why does this happen?

I assume the shower is on the 2nd story. Correct ?

Your shower and kitchen sink share the same drain. That drain is blocked. When you use shower, drain backs up and empties into the sink. Dishwasher is connected to the same drain either via its own drain or Garbage Disposer that's why it is effected as well.
Have your kitchen drain cleaned. Back to you. Milo

Btw: This would be the instance where Air Gap would do its job very well.

Milo is correct. You won't actually see the water in the sink because is stops at the sink trap. Immediatelt above the sink trap is a T connection for the dishwasher drain line which is much lower in elevation than other drains or traps. Water always takes the easiest route, in this case down to your washer. Remove the trap and horizontal pipe going into the wall. Rod into the pipe opening in the wall, use a small 1/4" rod because you muct make a 90 degree bend about 4" inside the wall. Look closely before removing the trap, sometimes you will find a clean out which will make life easier