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.