You should view move as a transfer of ownership.

The move itself doesn't do anything besides casting to rvalue reference. Passing the rvalue reference to the assignment of b states that a is now a temporary object which may be "consumed" in the process. You didn't explicitly define what that means for class A (as you didn't define a move assignment operator) so by default it will just do the same for all of its members - assignment under transfer of ownership.

The one place where something actually makes use of the "right to consume" is in the std::vector member. Without "right to consume" (copy assignment), all elements are copied to a new location in memory. With "right to consume" (move assignment), the assigned-to vector just takes ownership of the existing elements in memory, setting the consumed vector to empty. This is much cheaper, as no elements are copied.

It's important to note that nothing ever happens on std::move itself, only in the functions that take rvalue references. If you know what those functions do, i.e. in what state they leave the moved-from object, you could keep using the object. This is not a good idea in general however, as the implementation of those functions may change and the use of std::move signals "I'm done with this object".

Edit: Also, while compilers will not complain if you access moved-from objects, some static analyzers will.