Get closest point between GameObject (inside the cube collider) and 3D cube collider

The simplest (but computationally not the cheapest) would be to not rely on your current collider for the distance, but to add a set of small colliders around the edge of the object (so 6 colliders, one per face of the cube). Using Collider.ClosestPoint() on all 6 faces and calculating the distance like that would give you the results you need.

First convert a point to local space.

var localPoint = transform.InverseTransformPoint(worldPoint);
var extents = collider.size * 0.5f;
var closestPoint = localPoint;

Compute the distance to each face.

var disx = extents.x - Mathf.Abs(localPoint.x);
var disy = extents.y - Mathf.Abs(localPoint.y);
var disz = extents.z - Mathf.Abs(localPoint.z);

Find the closest face (smallest distance) and move the closest point along this axis.

if(disx < disy)
{
    if (disx < disz)
        closestPoint.x = extents.x * Mathf.Sign(localPoint.x); //disx
    else
        closestPoint.z = extents.z * Mathf.Sign(localPoint.z); //disz
}
else
{
    //......
}

Plus the offset of the collider, convert to world space.

closestPoint += collider.center;
transform.TransformPoint(closestPoint);

I don't know how efficient this is, but here is how I solved it:

    public static Vector3 ClosetPointOnBounds(Vector3 point, Bounds bounds)
    {
        Plane top = new Plane(Vector3.up, bounds.max);
        Plane bottom = new Plane(Vector3.down, bounds.min);

        Plane front = new Plane(Vector3.forward, bounds.max);
        Plane back = new Plane(Vector3.back, bounds.min);

        Plane right = new Plane(Vector3.right, bounds.max);
        Plane left = new Plane(Vector3.left, bounds.min);

        Vector3 topclose = top.ClosestPointOnPlane(point);
        Vector3 botclose = bottom.ClosestPointOnPlane(point);

        Vector3 frontclose = front.ClosestPointOnPlane(point);
        Vector3 backclose = back.ClosestPointOnPlane(point);

        Vector3 rightclose = right.ClosestPointOnPlane(point);
        Vector3 leftclose = left.ClosestPointOnPlane(point);

        Vector3 closest = point;
        float bestdist = float.MaxValue;
        foreach (Vector3 p in new Vector3[] { 
            topclose, botclose, frontclose, backclose, leftclose, rightclose 
        })
        {
            float dist = Vector3.Distance(p, point);
            if (dist < bestdist)
            {
                bestdist = dist;
                closest = p;
            }
        }

        return closest;
    }

(note: this assumes and axis-aligned box, which is all I needed at the time. If you want to rotate it you will have to do more work to transform the point.)

You can Calculate by Vector3.Distance

some example

 float minDistance =2;
 float Distance = Vector3.Distance(other.position, transform.position);
 if(Distance < minDistance)
 {
     //some code stuffs
 }
 else if(Distance > minDistance){
     //some code stuffs
 }

Useful information about Vector3.Distance and getting Distance from object source: https://docs.unity3d.com/ScriptReference/30_search.html?q=Distance