“If your question is whether State law requires you to obtain a marriage license if you get married under religious law, the answer is no.”
In New York it is illegal to perform any wedding without a license. You may be religiously married after a license-free chasuna but your Mesader Kedushin may be in legal trouble (misdemeanor with a $50 fine, and he loses his right to perform marriages for a while). There is also a 24 hour waiting period. Get the license. Both spouses have to be 18 in New York (16 with parental consent or 14 with parental and court consent — rarely given).
“Some (but not all) States will consider you legally married if you marry under religious law, and reside together with your spouse, even if you never get a State marriage license. “
This is correct, but New York is not one of those states. The term in secular law is “common law marriage”. New York forbade them in 1938, although if there is anyone still alive who entered a common law marriage prior to then the marriage is recognized. About a dozen states still permit them. And if you enter a common law marriage in another state and then move to New York, New York will accept it.