I don't know how timing is normally done for auto races, but I've been part of the staff at a few track & field events, such as the Nike Indoor Nationals, and seen how the timing is done there. There's a special camera that captures a single column of pixels, carefully aligned with the finish line, that records a rapid sequence of images whenever something changes in its field of view. The software assembles all these one-pixel-wide images into a single image whose horizontal axis is time rather than space. You can clearly see each runner, though arms and legs often look a little curved since they're moving as they pass through the plane of the finish line. In particular, the ID number tags worn by each runner are clearly visible.
Recognition isn't automatic, but it doesn't need to be. The operator just clicks on the front of each runner's shoulder (the part that "counts" for finishing the race) and reads the number from the runner's ID tag, and the time value associated with that column of pixels is recorded as that runner's finish time. So there's a human delay involved in matching the time measurements to the runners, but no human delay in the measurements themselves.
The company that did the timing at the races where I worked was CFPI Timing, and the tech page on their website has lots of details. Apparently the camera system I described is made by FinishLynx.