A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?
For a quantity to be a vector, magnitude and direction are not the only sufficient conditions. Rotation of a body is specified by direction. But, it is not necessary that it is always a vector quantity. Finite rotation of a body about an axis is not a vector because finite rotation does not obey the laws of vector additions. Only infinitesimally small rotation is a vector quantity.