How to detect that an object (ball or disk) stays in between the tile and on the border of a tile?

One possibility would be to add a collider to every tile when creating them, the collider being smaller than the tile. If the player ends his move outside all of the colliders, you can display a message, as he is no longer directly on any of the tiles

For that you would have to add the following lines to the code where you create each of those tiles at runtime:

 int x;   //  Play around with these values so that
 int y;   //  while the player is still in an acceptable
 int z;   //  position on the tile, he touches a collider with these dimensions

 //The next lines needs to be repeated on every created tile
 //so be sure to place it in a foreach loop or something similar
 Collider tempCollider = tempTile.AddComponent<Collider>();
 tempCollider.isTrigger = true;
 tempCollider.x = x;
 tempCollider.y = y;
 tempCollider.z = z;
 tempTile.AddComponent<BoundaryChecker>();

Here is the code for the BoundaryChecker.cs class:

 using UnityEngine;
 using System.Collections.Generic;
 
 public class BoundaryChecker: MonoBehaviour {
     private static List<BoundaryChecker> touchingColliders;
 
 
     void OnTriggerEnter(Collider other) {
         if(other.gameObject.tag == "Player")
             touchingColliders.Add(this);
     }
     void OnTriggerExit(Collider other) {
         if(other.gameObject.tag == "Player")
             touchingColliders.Remove(this);
     }
     public static bool isPlayerOnTiles() {
         if (touchingColliders.Count == 0)
             return false;
         else
             return true;
     }
 }

When the player ends his turn, you can call the static function BoundaryChecker.isPlayerOnTiles() which returns a boolean depending on whether the player is touching any of the colliders or not, and display a message accordingly.

Note:
Your player GameObject must have the tag "Player".