Timothy was one of Paul’s apostolic team, sent back to the church at Ephesus to address issues in that church while Paul was in prison. He carried Paul’s delegated authority. He was a timid personality, and this letter is full of encouragement from Paul. They are clearly very close to one another. It is a great example of how an older Christian might train and teach a younger one in a very relational way.
The gift of God is exactly that, a gift, and it is for Him to choose it, not the receiver. Having said that, we do have a responsibility to take action to ensure the gift is developed and properly used.