The parent's name doesn't affect insurance. Florida law does not state that to get a driver's license that a minor must be on their parent's auto insurance however likely the parent's insurance policy states that they must inform them of any licensed household drivers.
In Florida when a parent signs for a teenage child to get his driver's license with the FL HSMV they are already taking on liability and responsibility for the child's actions. FL state statute 322.09 is titled application of minors and responsibility for negligence or misconduct of minor. Subsection 2 states:
Any negligence or willful misconduct of a minor under the age of 18 years when driving a motor vehicle upon a highway shall be imputed to the person who has signed the application of such minor for a permit or license, which person shall be jointly and severally liable with such minor for any damages caused by such negligence or willful misconduct.
To get specific liable law regarding a parent's responsibilities and liability for signing the parental consent form and allowing a minor to obtain a driver's license contact the FL HSMV directly or a Florida lawyer that works in this area of law. The consumer division of the insurance regulator for the state of Florida may also be able to give you information on this topic in regards to what laws surround a minor and auto insurance.
We can tell you that FL insurance regulator notes on that any driver that resides in the home of the insured or has regular usage of the insured's vehicle, must be listed on the car insurance policy. If this information is not disclosed to the insurance company, they can deny a claim and cancel the policy for material misrepresentation (a form of insurance fraud).
If the driver is not living in the household and does not have regular usage of the vehicle and the insured provided permission to drive the vehicle, applicable coverage may apply.
So if the minor is living with the parents and obtains his driver's license it may not be required to of him to show proof of this to get the Florida license however the parent is stating they are taking on the liability and responsibility of the child as a driver when signing the parental consent form so the parents should want the child properly covered by auto insurance. Then there is the car insurance company that is covering the parent's car that will require the parents to notify them when a child becomes licensed so that he can be listed on the policy now as a driver.