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?

No, there are certain physical quantities which have both magnitude and direction, but they are not vectors as they do not follow the laws of vectors addition, which is essential for vectors.

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.