The simplest example would seem to be an egg of revolution, not an ellipsoid, with one point the North Pole at the pointy end, while the other is the pole at the broad end. The bisector is a circle of revolution, a parallel, but not an equator..
Note that here, if we take a plane containing the axis of revolution and intersect with the figure, the resulting meridian is a geodesic, while also being a bisector for any pair of points symmetric across the plane.
A less symmetric example starts with the xy plane in $\mathbf R^3, $ then introducing a radially symmetric hill with support within, say, the standard unit disk. The bisector of the points $(8,0)$ and $(10,0)$ is still the geodesic $x=9.$ However, the bisector of the points $(-2,0)$ and $(2,\frac{1}{2})$ is a little peculiar near the origin.
