ViewBag is a property of ControllerBase, which all controllers must inherit from. It’s a dynamic object, that’s why you can add new properties to it without getting compile time errors.
It’s not static, it’s a member of the object. During the request lifetime, the controller instance is created and disposed, so you won’t have “concurrency” problems, like overwriting the value.
The View (and its variants) method is not static as well, and this is how the view receives the ViewBag values: during the process of rendering the view, the controller instance has its ViewBag instance as well.