Can we know the region in the xy-plane given this information? Consider a closed smooth bounded curve enclosing a region S in the XY-plane. Suppose that we know the length of all lines which is the intersection of S with an infinite line parallell to the x-axis. And we know the same for the y-axis. Can we reconstruct S using only this data? 
If not, suppose we know the same for some other axes, is there a finite set of axis enough to reconstruct S?
No it is not possible to reconstruct such a shape. For instance, if you take a rectangle $A$ for which the sides form a $45°$ angle with the $XY$-axis, and rotate it by $90°$ to obtain a rectangle $B$, then the intersection of any line parallel to the x or y axis with $A$ will have the same length than the intersection with $B$.

For the generalization, I have no idea.
We know the function $f(x)$, where $x$ is a point on the $x$ axis, defined as the length of the intersection of, the line paralell to the $y$-axis which goes through $x$, with the set $S$, so it can have multiple components and we know the total length. I am owner of question