Why does applying a material to a object break batching?

Assigning a new material does not break batching. The object you assigned the new material will batch with other objects of that same new material. Here is a script you can run to verify. Attach it to an empty game object, drag and drop two or more materials into the materials array in the inspector, and hit run. Spacebar will advance to the next material. You should see that all the cubes are still batched even when the material is changed.

 #pragma strict
 var count = 100;
 var materials : Material[];
 private var iMat = 0;
 private var objects : GameObject[]; 
  
 function Start () 
 {
     objects = new GameObject[count];
     
     for (var i = 0; i < count; i++) {
         var go = GameObject.CreatePrimitive(PrimitiveType.Cube);
         go.transform.localScale = Vector3(0.25, 0.25, 0.25);
         go.transform.rotation = Random.rotation;
         go.transform.position = Random.insideUnitCircle * 5;
         go.renderer.material = materials[0];
         objects[i] = go;
     }
 }
  
 function Update()
 {
     if (Input.GetKeyDown(KeyCode.Space)) {
         iMat = (iMat + 1) % materials.Length;
         for (var i = 0; i < objects.Length; i++)
             objects[i].renderer.material = materials[iMat]; 
     }
 }

What breaks batching is changing some property of the a material like texture, or color, or blending, or offset, or... If you change a property of the material, Unity creates a new material instance...it is no longer the same material. The reason it breaks batching is that the shader needs all the properties to be the same in order for the game objects to batch. For example a shader cannot handle two materials where the main color attribute is different at the same time.