Grabbing objects for touch controls? how

You can use Input.GetTouch to get a Touch struct. One of the parts of this struct is the screen-space coordinate that was touched. You'll need a bit of extra code (see the linked documentation) for tracking specific fingers and so on, but that's the basics. You can then use Camera.ScreenToWorldPos to shoot a raycast into the world and see if the touch hit the hand. From there, it's just like dragging with a mouse - track the position to the touch until it is released, then stop tracking the position to the touch.

I forgot to mention about touch controls on desktop temporarily.. you should be able to just add code that would work on desktop and have it work on both.

EDIT: Added potential code solution as requested by question author.

 using UnityEngine;

 public class TouchTracking : MonoBehaviour {

     [SerializeField] private Transform target;
     [SerializeField] private Camera viewport;
     [SerializeField] private bool useMouse = true;

     // these keep track of important tracking info, such as finger and whether we're tracking or not
     private int trackingFinger = 0;
     private bool track = false;

     void Awake() {
         // if you leave the viewport field blank in the Inspector, 
         // use the first camera in your scene with the "MainCamera" tag.
         if (viewport == null)
             viewport = Camera.main;
     }

     void Update() {
         /**
         * Touch
         */
         int touchCount = Input.touchCount; // this tells you how many fingers are on the screen right now
         Touch touch; // this stores information about the touch
         for (int id = 0; id < touchCount; id++) {
             touch = Input.GetTouch(id);
             if (touch.phase == TouchPhase.Began && CheckCollision(touch.position))
                 track = true;
             else if (touch.phase == TouchPhase.Ended)
                 track = false;
         }

         /**
         * Mouse
         */
         // button 0 is left mouse
         if (Input.GetMouseButtonDown(0) && CheckCollision(Input.mousePosition))
             track = true;
         else if (Input.GetMouseButtonUp(0))
             track = false;
         /**
         * Perform the tracking operation if needed
         */
         if (track) {
             Vector3 inputPos;
             if (useMouse)
                 inputPos = Input.mousePosition;
             else
                 inputPos = Input.GetTouch(trackingFinger).position;

             Ray r = viewport.ScreenPointToRay(inputPos);
             float distance = (target.position - r.origin).magnitude;
             MoveToTouch(target, r.GetPoint(distance));
         }
     }

     bool CheckCollision(Vector3 position) {
         Ray ray = viewport.ScreenPointToRay(position);
         Vector3 worldPoint = ray.origin;
         Vector3 worldDirection = ray.direction;

         RaycastHit hit; // stores information about the raycast hit (if there was one)
         if (Physics.Raycast(worldPoint, worldDirection, out hit, Mathf.Infinity))
             return hit.collider.gameObject == target.gameObject;
         else return false;
     }

     void MoveToTouch(Transform target, Vector2 position) {
         float lerpedX = Mathf.Lerp(target.position.x, position.x, 100 * Time.deltaTime);
         float lerpedY = Mathf.Lerp(target.position.y, position.y, 100 * Time.deltaTime);

         target.position = new Vector3(lerpedX, lerpedY, target.position.z);
     }
 }