They measure kids with their shoes on, but they look at the shoes when the measurement is close and make a judgment call about whether the child is actually tall enough for the ride. They'll also, for example, gently put a hand on a child's head (or ask that a hat be taken off) to make sure that they are measuring the child and not the height of their hair, etc.
The measurement sticks/bars are pre-made and stand at each attraction with a height limit, and they serve the purpose of assessing safety but aren't necessarily to-the-millimeter accurate.
The best bet, and we went through this when our son was growing, is to say to your child that you'll just have to wait and see if he makes the height requirement(s) or not -- that only the CM at the ride can decide, and if the answer is "no" it is for safety's sake. We went through counseling him through tears when he wasn't tall enough and celebrations the following year when he was (turned out to be a great incentive to eat up those vegetables so he could grow!).