when you sync your controller to your Xbox, it will only respond to that Xbox afterward. The controller will not control or turn on your second Xbox.
If you want to have the controller control the second Xbox, you will have to hit the sync button on the second Xbox and pair it to the controller. However, when you go to sync to the second Xbox, since the controller is still paired to the first Xbox, you can turn on the first Xbox when you turn on the controller. So to avoid the issue, it's best to have dedicated controllers. Or don't switch the controllers too often between Xbox's.