Yes, the DMV requires auto liability insurance to register a vehicle in New York .
Okey, I have a bit of a problem: I have a different name on both my vehicle registration application and my Insurance Identification Card, do they have to be the same or that woudn't be a problem?
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?
Yes, they accept faxes or copies. Don't worry about that. However, if the barcode is not readable DMV won't accept it.
Thanks! One more question, I was living in a different State a few months ago, so I have an Insurance coverage from another state, If I want to register my vehicle here in new york can that out-of-state insurance be accepted?