A vector has both magnitude and direction. Does that mean 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?

The finite rotation of a body about an axis is not a vector because the finite rotations do not obey the laws of vector addition. However, the small rotation of a body {i.e. small angle of rotation) is a vector quantity as it obeys the law of vectors addition.