Let's go at a simpler question first, which will build up to yours. Suppose you have a single point and you want to know the length of the circle it traces out when you rotate this point around some axis. This obviously depends on the axis you choose. At one extreme, if the axis passes through the point itself, the point will trace out a circle of radius zero, which is zero length. On the other hand, if the axis is a $d$ meters away from the point at it's closest, the point will trace out a circle of radius $d$ meters and length $2\pi d$ meters.
Now, re-examine your question. Each point in $V$ is like the point in my example above and contributes a volume of $\sim2\pi d$ to the solid of revolution where $d$ is that point's distance to the axis of rotation. Clearly $d$ depends on the choice of axis, so the point's contribution to the total volume also depends on the choice of axis. To be specific to your question of the two axes being the $x$ and $y$ axes, only points on the line $x=y$ will be equidistant to both those axes. Most points in $V$ are different distances from the $x$ and $y$ axes. Unless $V$ were very symmetric (say, a circle centered at (1,1)), the total contributions from all it's points will be different depending on the axis chosen.