To adopt an adult in New York, the parent must submit an application to adopt. Upon receiving the application, the state of New York will.
DOH, Adoption Information Registry Birth Parent Registration Form, PDF. LDSS, NYS OCFS Request/Response for Name and/or address of Father of Child . 29a, Consent to Adult Adoption by Adoptee, PDF · Doc · WordPerfect.
There are many reason why people decide to adopt an adult, but there is essentially New York state carves out this exception in some limited.