Yes, ABS is at 1 and TCS is at 5 on every car you get into for the first time, which includes courtesy cars, quick match cars, and cars that are provided for a seasonal, even if they've been used before. Driving Line, Active Steering, and Skid Recovery Force are universal*, while Active Stability Management is off by default, and stays on for any car you have switched it on for.
*Skid Recovery Force is forced on in certain events, and certain other assists may be, too. Karts have no option for Active Steering, Active Stability Management, or ABS.