Yes, that's a problem. Your insurance cards and vehicles must have the same name.
I'll sort that out then. I want to know what sort of proof will I receive after getting the insurance?
Your insurance company will issue a proof of insurance in two ways: 1) They will either give you to original NY state Insurance Cards or provide you with acces to your digital electronic NY State Insurance.2) They will send  an electronic notice of insurance coverage  to the DMV.
Great! I accidentally lost my Insurance Identification Card but I have a copy, would DMV accept a copy or a fax of my IIC?