Yes, relative to the road (or someone observing from the ground), the tire is moving at 0 mph at the point it touches the ground and moving twice the speed of the car, 120 mph, at the highest point of the tire.
Relative to the driver of the car, the “rim” of the tire is moving at the speed of the car, 60 mph. Any other point on the tire has the velocity v r / R, where r is the distance between the point and R is the total radius of the tire.
However, relative to a meteoroid, a biker on the road, a car going in the opposite direction, etc. all these values are different. There is a level of ambiguity when the question “What is the speed of . . . ?” is asked without stating the observer.