$A$ 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?

(N/A) No; No.
$A$ physical quantity having both magnitude and direction is not necessarily a vector. For example,electric current has both magnitude and direction but is a scalar quantity because it does not follow the laws of vector addition.
Similarly,the rotation of a body about an axis is specified by the direction of the axis and the angle of rotation. However,finite rotations do not follow the commutative law of vector addition (i.e.,$\vec{A} + \vec{B} \neq \vec{B} + \vec{A}$ for finite rotations). Therefore,finite rotations are not vectors. Only infinitesimal rotations follow the laws of vector addition and are considered vectors.