As for the New York D.M.V., a spokeswoman, Lisa Koumjian, said the agency was finding its way.

“The state has taken a number of affirmative steps to protect and advance the civil rights of all New Yorkers, including those who are transgender or gender nonconforming,” she wrote in an email. “We expect to work with the city and the different systems to support this new initiative.”

A State Senate bill to allow “X” on driver’s licenses stalled in committee last year.

New York State’s Department of Health, which studied the implications of nonbinary birth certificates, said it was concerned that they could cause people to be denied Medicaid benefits because the federal and state data sources used by Medicaid to verify eligibility recognize only male and female.

The new New York City birth certificate law, sponsored by the Council speaker, Corey Johnson, also lets people change the sex on their birth certificates without a note from a health professional, a change that transgender advocates had long pushed for.

Mx. Arrowood conceded that their gender identity can be difficult for others to understand. Mx. Arrowood’s birth certificate, issued at Lenox Hill Hospital in Manhattan 31 years ago, said female.

“I was never transitioning ‘towards male,’” Mx. Arrowood said, “but away from female.”

(Mx. Furuya said it often takes them a few tries to explain who they are, too. When asked how they identified, they sent a diagram with four category umbrellas and nine subcategories.)

Gretchen Van Wye, assistant commissioner in the city health department’s Bureau of Vital Statistics, said that with the new law, the city was struggling to navigate the space between “sex,” the biological category that a birth certificate purports to define, and gender, which is largely seen as more subjective.