Does a flight training company need to make a copy of my driver's license?
According to 49 CFR 1552.3 a flight school is required to keep a copy of the documents used to provide proof of citizenship for five years. This is usually a passport or birth certificate. A driver's licenses is not proof of citizenship in the US. 

However, a valid photo ID (such as a driver's license) is required in addition to your pilot's license in order to exercise the privileges of the pilot's certificate because the FAA issued pilot's licenses do not have a photo for identification. So a business that is renting an aircraft to you has a legitimate interest in retaining evidence that they are handing over their valuable asset to a licensed pilot, and that you are who you say you are.

They may not be required to keep a copy of your driver's license, but they are also not required to rent you an aircraft or provide flight instruction.