it is possible for objects to orbit multiple objects at the same time. Add the moon to the system. The moon is orbiting the earth that is orbiting the sun that is orbiting the center of our galaxy. And yes each of them have a common center, just that it is very very close to the center of mass of the larger object in each case.

For the moon the earth is the dominant gravitational force, for the earth it is the sun and for the sun it is the center of our galaxy

There's a very famous smart man with zany hair and a big tongue that says your comment is wrong and it's all relative to your frame of reference.

No the common center is called a barycenter and it’s somewhere just outside the middle of the sun.