I will only discuss rigid bodies here; I do not understand fluids well and I doubt you were thinking about fluid bodies anyway.
The first thing you need to understand is that the concept of "moment of inertia" can best be understood in terms of the inertia tensor. The word tensor might deter you, but you must face it to understand many concepts in physics. For pedagogical purposes the essential idea of tensors is that what you are interested in should not depend on the coordinate system you use. You suggest computing moments of inertia about many different axes, but this is the same as computing moments of inertia in different coordinate systems. Since the moments of inertia are part of a tensor we do not gain information by computing them in different coordinate systems. A tensor computed in one coordinate system is equivalent to that tensor computed in a different coordinate system.
In particular, the inertia tensor contains only three independent numbers. The easiest way to obtain these three numbers is to compute the moment of inertia about the three principal axes (let me call these the $x$, $y$ and $z$-axes) of the rigid body. Then the inertia tensor is fully specified by the three numbers $I_x$, $I_y$ and $I_z$. If you now compute the moments of inertia about new axes, $x'$, $y'$ and $z'$, I can tell use the new moments of inertia in terms of the principal moments $I_x$, $I_y$ and $I_z$. In other words, computing the moments of inertia in different coordinate systems does not add information.
One interesting consequence is that any rigid body can be modeled as an ellipsoid with uniform density, as long as we only care about rotation and translation. This is because an ellipsoid can be made to have an arbitrary inertia tensor ($I_x$, $I_y$ and $I_z$) by changing the shape of the ellipsoid.
TL;DR The moments of inertia of a uniform density solid are completely determined by only three components, so we cannot determine the shape of a body from moments of inertia alone.