Find the normal between the two then scale that normal by the distance wanted.

float distanceWanted = 3.0f;

Vector3 diff = transform.position - player.position;
diff.y = 0; // ignore Y
transform.position = player.position + diff.normalized * distanceWanted;

Find the normal between the two then scale that normal by the distance wanted.

float distanceWanted = 3.0f;

Vector3 diff = transform.position - player.position;
diff.y = 0.0f; // ignore Y (as requested in question)
transform.position = player.position + diff.normalized * distanceWanted;

This will also keep the y of your following object to be exactly the same as player.position, you may want to add a vertical offset relative to the player:

transform.position = new Vector3(0.0f, verticalOffset, 0.0f) + player.position + diff.normalized * distanceWanted;

Find the normal between the two then scale that normal by the distance wanted.

float distanceWanted = 3.0f;

Vector3 diff = player.position - transform.position;
diff.y = 0; // ignore Y
transform.position = player.position + diff.normalized * distanceWanted;

Find the normal between the two then scale that normal by the distance wanted.

float distanceWanted = 3.0f;

Vector3 diff = transform.position - player.position;
diff.y = 0; // ignore Y
transform.position = player.position + diff.normalized * distanceWanted;