Exact same issue here (PC Player)
When I first launched the game I played a couple of hours and had no problems whatsoever with customization. After I relaunched the game I suddenly had a different car than the one I originally chose, a different character and 0$, meaning I lost about 3 hours of progression. Only thing that hadn't changed was my rep level. (joy!)
Since then the server hasn't been able to "retrieve your wraps" either. I also can not download "community wraps" as it says that my "slots are full" (haven't downloaded a single wrap btw, and only customized 1 myself). So there is no way I can wrap my cars as of right now, which is about 50% the fun of the whole effing game.
Found several threads dating back since 2015 of other NFS games from people with the exact same issue. Only thing EA can come up with is connection / port forwarding issues, which I know for a fact isn't the case since I got that all sorted. Knowing EA they will just ignore this issue, again.
For some people it worked after relaunching their game or after restarting their pc. Not for me though.