Your question seems to be based on some false assumptions.
As far as I can tell:
There is no federal requirement for an Uber driver to have a class A, B or C license.
Just because they are commonly called "commercial" licenses, it does not follow that every kind of commercial driving necessarily requires you to have one.
The US Department of Transportation has a website describing what kinds of driving require these license types.
Class A: Any combination of vehicles which has a gross combination weight rating or gross combination weight of 11,794 kilograms or more (26,001 pounds or more) whichever is greater, inclusive of a towed unit(s) with a gross vehicle weight rating or gross vehicle weight of more than 4,536 kilograms (10,000 pounds) whichever is greater.
Class B: Any single vehicle which has a gross vehicle weight rating or gross vehicle weight of 11,794 or more kilograms (26,001 pounds or more), or any such vehicle towing a vehicle with a gross vehicle weight rating or gross vehicle weight that does not exceed 4,536 kilograms (10,000 pounds).
Class C: Any single vehicle, or combination of vehicles, that does not meet the definition of Class A or Class B, but is either designed to transport 16 or more passengers, including the driver, or is transporting material that has been designated as hazardous under 49 U.S.C. 5103 and is required to be placarded under subpart F of 49 CFR Part 172 or is transporting any quantity of a material listed as a select agent or toxin in 42 CFR Part 73.
A typical Uber driver operating a typical passenger car (say, 5 passengers and 4000 pounds) would not fall into any of these categories.
Some states do require an Uber driver to have a special kind of license; others do not.
For example, New York has a Class E license, which for-hire drivers are required to have. (Ordinary people typically get a Class D license instead.)
On the other hand, consider Colorado. You can read about Colorado license classes here (PDF). Most ordinary people get a Class R license, and there is no indication that driving a passenger car for hire requires anything other than this.
In states where such requirements do exist, it appears that Uber verifies that its drivers do indeed have the appropriate license.
For instance, here is Uber's page for prospective New York City drivers. It explains that you have to get a Taxi & Limousine Commission (TLC) license, as well as a Class E driver's license.