Would different car insurance or health and safety analysis be needed for driving supercar on road?

US insurance rules are set by states and they differ somewhat. There used to be a few states that did not require insurance but I don’t know if that is still true. Also if you have a car loan the bank or whoever gives the loan will require insurance.

I think it may along the lines of - if the daughter is not listed on the insurance policy and has an accident, then the insurance company will make good the damage but then chase her to re-reimburse them. It depends what insurance she has by herself and what it says about borrowed cars. A lot less hassle if the driver is listed on the insurance. And as pointed out, every state has its own peculiarities.