Its not a standard practice per se, but is certainly more and more commonly used at the larger airfileds ( speaking purely from a UK perspecitve ). Depends on each airfileds individual procedures.
Ref the a/c type being on radar. Generally speaking the a/c type wont be automatically displayed on each aircrafts SSR label, but it will be printed/written on the flight progress strip. I would say that reporting the a/c type serves as;
1 - a reminder to the controller, similar to stating "(callsing) Heavy" on first contact.
2 - to confirm what is presented to the controller on the strip display, becasue the a/c type may have been incorrect in the FPL.