What happens if it is register under a different but the insurance is under my name

As long as the car is registered, an has insurance there should not be a problem. Technically it's not their place to check for that. Their job is to make sure the vehicle is safe for the road

Not sure if they will reject it or not, but your registration and I sure will usually have something in common, like address and VIN. See https://www.usinsuranceagents.com/answers/does-name-insurance-card-jersey-have-match-cars-registration/