Nope, that is an original "Mini" Mini.
(Also known as BMC Mini, Morris Mini, Austin Mini and Rover Mini (and probably a few more too!) throughout it's life.)
That car was available in certain versions of GT4, (NTSC/Japan I believe.) but not in NTSC U/C or PAL.
When BMW designed the "Bini" (It's not a real Mini!) it was trying to make the car look as much like the original as possible whilst meeting modern day safety & practicality requirements.
If anything the 2000 BMW MINI set a retro-trend which Ford followed in 2002 with the 11th Gen T-bird and 2005 with the 5th Gen Mustang, while Dodge did likewise with the 2006 Charger and more recent Challenger.
There were probably other retro-examples before BMW's commercial offering, but I wouldn't say BMW followed Ford or Dodge's cues.