In Player Career, when a driver, real or custom, moves teams, their name is only changed outside the garage, but inside on the walls(eg. Aston Martin), the original driver name and number is still there.
Additionally, teams like Ferrari have their driver names on the back of their suits but is not there when drivers move.
Red bull drivers have their name and flag on their car but when new drivers join(or at least created drivers) it's blank.