It is incorrect to talk about "The" bisector of an ellipsoid, unlike a sphere it doesn't have just one that is the same throughout. The bisectors would be ellipses, but I am not aware of any special names given to one of them.
1. In general a bisector is one dimension lower than the bisected object. For example:
the bisector of a circle (dim = 2) is a straight line (dim = 1)
the bisector of a solid (dim = 3) is a plane (dim = 2)
2. There are 2 different types of ellipsoids:
a) Elliptic ellipsoid: Then every bisector is an ellipse
b) Rotational ellipsoid: Then the bisector is either a circle (the axis of rotation is perpendicular to this circle) or an ellipse.
EDIT: After playing around a little bit I noticed that it must be possible to get a circular bisector with elliptic ellipsoids too!