Answer
No, it is not possible for an object to have the same moment of inertia for all possible axes.
Yes, it is possible for an object to have the same moment of inertia for all axes passing through a certain point.
Work Step by Step
The parallel-axis theorem (page 288) relates the moments of inertia for an object when it is rotated about 2 axes that are parallel. When a rotation axis is moved parallel to itself, there is always a non-zero change in the moment of inertia, as we see from the second term in equation 9.19. Therefore, it is not possible for an object to have the same moment of inertia for all possible axes, except perhaps for the trivial case of a point particle where only one axis is possible.
It is possible for an object to have the same moment of inertia for all axes passing through a certain point. An example is that of a solid uniform sphere, and the point is the center of mass (i.e., the sphere’s center). For all such axes passing through the center, for any orientation, the moment of inertia is the same: $\frac{2}{5}MR^2$.
