I don't think Hausdorff distance is an appropriate measure of "contact".
Imagine we have two objects, finite in extent, touching at one point, with neither fully enclosed by the other. I can increase the Hausdorff distance to be arbitrarily great by taking part of one object's surface on the far side from the contact point, and extruding it into a spike or tendril that escapes to a great distance from the other object. I haven't changed the nature of the contact, I've just changed the shape of a part of the object nowhere near the contact, and yet I've completely distorted the measure.
So I think what you want is something more like the closest distance / penetration distance, or a measure of the volume of space sandwiched "between" the objects.
If your objects are, or can be well approximated by, collision primitives, then you may be able to use the physics engine to estimate the smallest distance between them:
If a collider of one object intersects a collider of the other, Physics.ComputePenetration will tell you how deeply it penetrates (ie. what is the minimal translation vector we'd need to apply to separate them?)
If none of the first object's colliders intersect any of the other's, then you can use
Physics.ClosestPoint to find the closest separations between them. If your shapes are convex, then it suffices to find the closest point on each collider to the center of each of the other object's colliders, then finding the closest point on the other object's collider to that point. If your shapes can be concave, then you may be forced to scatter sampling points along the surface of the concave shape and try to find closest points to those samples.
Another strategy you may be able to use if your contacts are "plane-ish" (ie. not wrapped around, like entwined chain links) is to use the rendering system to convert the 3D space to a raster image you can more easily analyze.
First, identify the approximate axis of approach - this might be as simple as a line drawn through each object's centres, or determined using crude collider approximations as above.
Position an orthographic camera so that it looks along this axis, scaled so at least one of the objects just fills its view (it's OK if the other extends out of frame - since we're looking for contacts, we only need the pixels that overlap both objects)
Render the further object's front faces, with depth-testing set to "less", drawing the depth of each rendered pixel to one render target.
Render the nearer object's back faces, with depth-testing set to "greater", drawing the depth of each rendered pixel to a second render target.
Subtract one render target from the other: this is the depth of separation between the objects at this point, along your chosen axis. It will be negative where the objects intersect (a penetration depth).
Now you can apply any image analysis technique you can imagine to this separation depth map. You could look for its local minima (near-contact points or deepest intersections). You could sum areas of pixels to compute the volume of space in the sandwich. You could form a histogram of separation distances to categorize things that approach at only a few points vs objects that stay closely parallel/lapped over a large area...