(2D) Make a GameObject Shake using script in Unity

I did this code now, I think it might help:

     IEnumerator Tremble() {
           for ( int i = 0; i < 10; i++)
           {
               transform.localPosition += new Vector3(5f, 0, 0);
               yield return new WaitForSeconds(0.01f);
               transform.localPosition -= new Vector3(5f, 0, 0);
               yield return new WaitForSeconds(0.01f);
           }
     }

The idea is to change directly the localPosition very fast to different directions. Check if it is what you are looking for or if you can work on it to get there

Maybe something like this? I left much of your normal fall code alone, just coroutined it.

The idea is to have a sin wave drive the shake, so it's a bit of a smooth shake.

public class FallingBox : MonoBehaviour
{
    [SerializeField] float fallSpeed = 3f;
    [SerializeField] Transform target;
    
    Rigidbody2D rigidbody;
    BoxCollider2D boxCollider;

    void Start()
    {
        rigidbody = GetComponent<Rigidbody2D>();
        boxCollider = GetComponent<BoxCollider2D>();
    }

    //Called when entering the "danger zone"
    public void ChangeStatus()
    {
        StartCoroutine(OnStartFall());
    }

    IEnumerator OnStartFall()
    {
        var maxTrembleTime = 0.25f;
        var currentTime = 0.0f;

        while(currentTime < maxTrembleTime)
        {
            currentTime = Mathf.Clamp(currentTime + Time.deltaTime, 0, maxTrembleTime);

            // sinOffset = 0 will have us back at the same point.
            // You might want to scale this number by a constant to make it shake faster or slower
            var sinOffset = (maxTrembleTime - currentTime); 
                                                
            var offset = Mathf.Sin(sinOffset); // Scale this offset to make it shake over more distance
            transform.position.y += offset;

            yield return null; // wait until next frame
        }

        // Begin normal falling code

        while(true)
        {
            var frameMovement = fallSpeed * Time.deltaTime;
            var targetPosition = target.transform.position;
            transform.position = Vector2.MoveTowards(transform.position, targetPosition, frameMovement);

            if (transform.position == targetPosition)
            {
                gameObject.layer = LayerMask.NameToLayer("Ground");
                boxCollider.enabled = true;
                break;
            }

            yield return null; // wait until next frame.
        }
    }
}

I'm not a huge fan of coroutines in most situations, when simple logic will do the trick.

Here's an example of just using a Box State to determine what the box should be doing at any given time:

public enum BoxState
{
    None,
    Shaking,
    Falling,
    Finished
}

public class FallingBox : MonoBehaviour
{
    [SerializeField] float fallSpeed = 3f;
    [SerializeField] Transform target;

    [SerializeField] float shakeTime = 0.5f;
    [SerializeField] float shakeSpeed = 1f;
    [SerializeField] float shakeFactor = 0.1f;

    Rigidbody2D rigidbody;
    BoxCollider2D boxCollider;

    private Vector3 originalPosition;
    private Vector3 targetPosition;
    private float timer;

    public BoxState boxState { get; private set; } = BoxState.None;

    void Start ( )
    {
        rigidbody = GetComponent<Rigidbody2D> ( );
        boxCollider = GetComponent<BoxCollider2D> ( );
    }


    void Update ( )
    {
        switch ( boxState )
        {
            case BoxState.None:
            case BoxState.Finished:
                return;

            case BoxState.Shaking:
                // Use Perlin Noise here for a quasi-random shake. Multiply or Add extra values for noise speed variations.
                // Many different ways to produc these offset values. This is just one variation.
                var xOffset = Mathf.PerlinNoise ( Time.time * shakeSpeed, 0 );
                var yOffset = Mathf.PerlinNoise ( 0, Time.time * shakeSpeed );

                transform.position = originalPosition + new Vector3 ( xOffset, yOffset, 0 ) * shakeFactor;
                timer += Time.deltaTime;
                if ( timer > shakeTime )
                {
                    boxState = BoxState.Falling;
                    timer = 0;
                }
                break;

            case BoxState.Falling:
                transform.position = Vector2.MoveTowards ( transform.position, targetPosition, fallSpeed * Time.deltaTime );
                if ( ( transform.position - targetPosition ).sqrMagnitude < Vector3.kEpsilon )
                {
                    transform.position = targetPosition;
                    gameObject.layer = LayerMask.NameToLayer ( "Ground" );
                    boxCollider.enabled = true;
                    boxState = BoxState.Finished;
                }
                break;            
        }
    }


    //Called when entering the "danger zone"
    public void ChangeStatus ( )
    {
        if ( boxState == BoxState.None )
        {
            boxState = BoxState.Shaking;
            originalPosition = transform.position;
            targetPosition = target.transform.position;
            timer = 0;
        }
    }
}

I also personally find that a solution like this is much more easily updatable. Adding an extra state is far easier here, than creating and timing new coroutines.