What is the general class of manifolds isometrically embedded in $R^3$ for which there exists some point in $R^3$ such that any plane through the point yields a geodesic where it intersects the manifold? Two immediate examples are: a plane, with respect to any point; a sphere, with respect to its center point.



Class of embedded 2D manifolds whose geodesics are coplanar with a point
Only spheres and planes have this property.


I assume that the surface is sufficiently smooth ($C^2$).
Every spatial curve has the so-called principal normal at every point of non-zero curvature, this is the acceleration direction for the unit speed motion along the curve.
The principal normal to a geodesic coincides with the normal to the surface. Now take any non-flat point $X$ (that is with at least one principal curvature non-zero) on your surface and draw all planes through your "center point" $O$ and $X$. Most of them will intersect the surface along a curve with non-zero curvature at $X$. Since the normal to a curve contained in a plane also lies in the same plane, it follows that the normal to the surface at $X$ lies in most planes through $O$ and $X$, and hence it lies on $OX$. Conclusion: at every non-flat point $X$ the tangent plane is perpendicular to $OX$.


It follows that each connected component of the set of non-flat points is contained in a sphere. Hence the set of non-flat points is closed. But it is also open. So, if there are no non-flat points, the surface is contained in a sphere. And if all points are flat, then the surface is contained in a plane.


