Mbappe will get privilege base on his performance and contribution to the Madrid's team but isn't something that will happen immediately but with time and base on his contribution in the club, you can take this part away, basically the fact that when these big name players joined a particular club most of them usually get special treatment from their manager and also further having more influence or say in the team than every other player in their.
I don't think their is any other privilege for him than to play good food ball when he plays good football then he will get a lot of acknowledgement. And he will will have his dream come through which is to win the Champions League and his performance can also earn him personal trophies. I don't think Madrid will give special treatment, and if eventually he gets special treatment then lucky him. I don't what he needs are beyond what he thinks, this is a opportunity for him to be the best with the kinds of team mates he going to have, so he needs to work with them to be able to get his dream accomplished, and he as Vinicius Junior to support him, so I feel Mbappe will even have more supporters from Madrid than he had in psg.
Currently I think Mbappe is about to get one of his biggest achievements which is to be able to win the world best player, to win the champions league which he couldn't win with PSG and to be seen as the best when it comes to individual achievements. Mbappe has got himself an honor by choosing to play for Real Madrid I believe he already have the privilege to play alongside with the biggest and best young players in the world right now in the person of vinicius junior, Jude Bellingham and others. Mbappe having the best manager in the world right now as his coach is really a privilege because he will improve his playing abilities and improve his performance as a professional footballer so his performance and relationships with his teammates will determine how lucky he's going to be in the team.