Many 3d authoring applications support some shape or form of this. Some refer to the technique of creating shapes by carving them with other shapes as a "Boolean Operation". There's a nice acedemic term for this, but I can't seem to remember it right now.

Nothing is stopping you from programming this in Unity, however it's pretty heavy on math, so you'd probably not want to start without a cup of coffee on your desk.

Have you looked into creating a mesh with lots of vertices and then altering the mesh?

You'll need to do a camera.ScreenPointToRay() to find which triangle you are clicking on, then using the triangle you can find it's 3 points and then alter the mesh accordingly.

You'd need to compensate for the rotation of the object etc.

Example from Unity Reference Manual

function Update () {

  // Only if we hit something, do we continue

  var hit : RaycastHit;

  if (!Physics.Raycast (camera.ScreenPointToRay(Input.mousePosition), hit))

    return;

      // Just in case, also make sure the collider also has a renderer material and texture

      var meshCollider = hit.collider as MeshCollider;

      if (meshCollider == null || meshCollider.sharedMesh == null)
    return;

    var mesh : Mesh = meshCollider.sharedMesh;

    var vertices = mesh.vertices;

    var triangles = mesh.triangles;

    // Extract local space vertices that were hit

    var p0 = vertices[triangles[hit.triangleIndex * 3 + 0]];

    var p1 = vertices[triangles[hit.triangleIndex * 3 + 1]];

    var p2 = vertices[triangles[hit.triangleIndex * 3 + 2]];


    // Transform local space vertices to world space

    var hitTransform : Transform = hit.collider.transform;

    p0 = hitTransform.TransformPoint(p0);

    p1 = hitTransform.TransformPoint(p1);

    p2 = hitTransform.TransformPoint(p2);

    // Display with Debug.DrawLine

    Debug.DrawLine(p0, p1);

    Debug.DrawLine(p1, p2);

    Debug.DrawLine(p2, p0);

}