There are many characteristics of New York that make it a fascist state, but this isn't one of them. For all its flaws, New York has a reputation for having one of the most effectively regulated insurance industries in the nation -- which is why insurance companies often brag about their New York presence as a symbol of their stability.
The situation you describe isn't a symptom of a fascist state. It's a symptom of a state where insurance fraud is a major problem -- largely because it's filled with quasi-humanoid mutants who have no business living in a free, orderly society. They've figured out that an easy way to drive without insurance is to make one initial payment on an annual policy, get the insurance card to verify that they are insured, and then drive around for the rest of the year without making any more payments. The insurance company will cancel the policy, but that doesn't stop them from driving around and putting other people at risk without any insurance coverage.
Having a paper insurance card still ‘proves’ you have insurance in NY state? How backward! Cops here check the database. Having an insurance card means nothing.