I think generally any kind of organic shapes would work well, such as limbs, e.g. an ellipsoid for an arm, one for the head, etc. I think the main use of it would be if you had say hundreds of rag-dolls then you would save a lot of time since you could model them using fewer ellipsoids than other shapes.
Or for modelling hit areas, it would be a nice shape to use. Maybe just for static volumes.
I guess, if you wanted to add a really simple collision volume to a flying saucer…
It’s just an idea! Something to think about.
I think you may be right that it’s one of those things that would be really nice (and you could probably make some really efficient games and reduce frame-rates and have more characters on-screen) but the potential bugs it could introduce might out-weigh the benefits. And if it didn’t go through the physics engine then you might not get the benefit. IDK Maybe there’s a reason why PhysX doesn’t support it. Too much trouble!
Maybe one of the devs who likes maths might implement it as a add-on just for fun! 
I forgot to add: If you put a sphere collider on a sphere, then stretch or rotate that sphere, the collider no longer is the same shape as the object (if you don’t have ellipsoids). Whereas for a box collider it just becomes a cuboid.
So basically the need for it would be to be able to stretch game objects which have cube and sphere colliders on them. Currently that would only work for cuboid, and mesh colliders.
It might work like this:
You’d have to work out the perpendicular to the plane of intersection of the two ellipses and the mid point of the intersection. And also how far in the ellipses overlapped. Unlike the sphere case there would be a torque on the ellipsoids as they hit each other.
The equations of a general rotated ellipsoid can be expressed as Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+1=0. (So 9 numbers compared to a spheres 4).