Your friend's case is unusual, maybe the result of something like obsessive compulsive disorder. In general, kids are much more resilient and adaptable than adults usually give them credit for. When you change something on them, they explore their boundaries under the new set of circumstances in an almost methodical way if you stop to observe. This can be stressful while they figure out their boundaries and where they fit in, but once they figure it out, they are fine. They then know how to adjust their behavior to fit the circumstances. This takes longer for some children than others, but they eventually all get there.
Where problems come in is when the circumstances are seemingly identical to the child, but expectations change. If Grandma lets him raid the cookie jar freely, but Mom gets angry when she catches him, he will be confused at first, but quickly figure out he needs to add the variable of who is in charge to his mental model, and he adapts easily to the change. However, if Mom punishes him harshly sometimes and lets it slide other times with no explanation, he has a difficult time processing it. Likewise, a big change like moving is traumatic, but kids eventually adjust and move on with their lives. However, if a child never knows when he will be moving again with little to no notice, he might develop some unhealthy coping strategies to try to maintain stability, like never trying to make friends.
In other words, when people talk about kids needing stability, they don't mean never changing anything, they mean when the inputs the child can observe and comprehend are the same, he can depend on the outputs being the same.