What does it mean to have sole ownership of object for unique_ptr?

Ownership is all about: who cleans up the resource when it is no longer needed?

To be the only owner of a resource means that only you are responsible for deleting the object when it is no longer needed. This implies that nobody else can use the object after your lifetime ended. If others depend on the object still being alive after your lifetime ended you need shared ownership.

In modern C++, raw pointers should not participate in ownership. When you pass a raw pointer to a function you expect that this function will not hold on to this pointer to use it later.

Ownership talks about how is responsible about the lifetime of the object.

Let say you have a class shape that on which you can do various actions like, calculating the surface area, enlarging or shrinking that object, ...

And those tasks are performed by free functions calculate_shape_area, enlarge_shape, shrink_shape, and you pass the shape to that function, then those functions don't require ownership over that object, they only need to know the object while they are called. So you can (and should) pass that object as raw pointer:

struct shape {};

double calculate_shape_area(shape* obj) {
   // calculate the area, obj is only used within that function
   // and not stored anywhere else
}

int main() {
  auto obj = std::make_unique<shape>();
  auto area = calculate_shape_area(obj.get());
}

If you, on the other hand, want to store that shape in a scene graph using insert and that scenegraph, at a later step, displays all shapes. Then the scenegraph needs to know about that object after the call of insert so it either needs to create a copy of it or needs to claim ownership over it. And if you don't want to create a copy then you either need to move a unique_ptr to that insert or use shared_ptr.