Is the amount that the company exempted a gift to an employee who provided a service to a friend rightfully his, or rightfully the friend's who received the service?

If you did not inform the company that the service is for your friend, then you must inform them. And if you informed them and they took a fee for the service, then it is their right. If they return the money, then inform your friend that he is exempted from the fees because you are an employee, and you are not entitled to take anything from it. If your friend waives the amount or a part of it to you, then you may take it; otherwise, you have no right to it.