1. Whether one is married according to our law, is solely a function of halacha (e.g. kesubah, kiddushin, hupah, etc.).
2. While in many (probably, most) states, including New York, a marriage license is not required (i.e. if you marry without one your halachic ceremony constitutes a valid civil marriage, meaning you need a civil divorce to marry anyone else, and you are committing fraud if you claim to be unmarried), it is very inconvenient not to have a marriage license. The alternative would be to bring in affidavits and other evidence for every transaction that requires proof of marriage. Note that this frequently is an issue only when a spouse dies (i.e. is the woman his widow and mother of his children, or a single mother whose children paternity is unknown who happened to be his live-in “best friend”).