Street car would have DOT approved tires, emissions, passing state and/ or local guidelines and restrictions with a registered VIN and license plate. You can get custom built car registration as well it’s just more work.
A race car would need to meet NHRA, IHRA, NASA, or any other automotive class requirements for safety and class specific regulations, but not necessarily DOT requirements as the vehicle is not intended to be driven on the street.
I would draw the line at the point where you need to tow it to the track to race in a street class.