NASCAR drivers do count as professional athletes. Legally speaking, a professional athlete is a person who performs services in a professional athletic event in exchange for money or other compensation. NASCAR drivers do perform the service of racing their car for their racing team in a professional athletic event, and they receive compensation for it, so they fit that definition.
There are some people who argue that NASCAR racing is not a sport, therefore NASCAR drivers are not real athletes. This is not true. Sports are athletic activities that require skill or physical prowess, usually of a competitive nature, such as racing, baseball, football, tennis, golf, bowling, wrestling, boxing, hunting, fishing, etc. One of the primary characteristics of a sport is competition. NASCAR is certainly of a competitive nature, with each racer trying their best to beat out the rest of the drivers and win the race. They spend years practicing for races, and give everything they have to try to win when it comes to race day.
Another indicator of a sport is the skill or physical prowess required to compete in it. NASCAR drivers must race at full speed and have to continuously turn sharply around tight corners, while coming dangerously close to dozens of other fast moving vehicles. This requires intense focus and incredible fine motor skills and reflexes. Many NASCAR races take 3 hours, with some of the longer ones even lasting more than 4 hours. Racers maintain a speed of more than 200 miles per hour for the entire duration of these races, and they are constantly turning, swerving, breaking, accelerating, so their arms and legs are engaged for multiple hours on end. Because of this, NASCAR drivers must have top tier physical endurance, both in muscle and cardio. This, combined with the need for unending focus, makes the sport unique in that it can be both mentally and physically exhausting. Their only chances to rest come with their pit stops, which rarely take more than 15 seconds, and only happen a few times per race.
While not all sports involve it, many people consider teamwork to be another indicator of a sport, and NASCAR has it. When it comes to pit stops, it is vital that each member of the pit crew performs their task as efficiently as possible with no mistakes, or they risk losing the race. It is not just up to the driver. The team has to build a car that sets them up for success. The crew chief must communicate with the driver and make strategy decisions throughout the race. Races can be won by milliseconds, so each decision, pit stop, and team member can have a significant impact on the result of the race.
It may not involve a ball like most mainstream sports, nor physical exertion in the way most people picture it (running, jumping, etc.), but this does not exclude NASCAR from the definition of a sport. Since NASCAR drivers compete in a sport, and are paid for the services in doing so, they do count as professional athletes.