Not only in circular motion, but in entire kinematics we discuss the case of point objects only. In kinematics we study motion only and not the objects undergoing the motion.
A point object is an object having negligible dimensions compared to the distance travelled by it. For example, a car moving from Delhi to Mumbai can be considered a point object.
Further, when we discuss the concepts in Kinematics, even a large object is dealt considering the entire body to be a single point. This point will be the centre of mass of the body.
If a person is revolving a stone tied to a string, the centre of mass of the stone is taken for determining the length starting from the point about which it revolves.
In dynamics, we have to consider the shape and size of the body also.