Some history in regards to the situation in New Mexico right now.
In November 1972 during/immediately after the Federal Equal Rights Amendment was submitted to the States, the voters of New Mexico passed a state level version of the ERA (went into effect in July 1973). In the aftermath of this, the Legislature of New Mexico rewrote the laws of the state to make them gender neutral (everything from inheritance to divorce law). This included dropping the explicit requirement in Marriage Law that marriage was a man and a woman. *However*, one of the other parts of the law (section 40-1-17) is that "the form of application, license and certificate shall be substantially as provided in Section 40-1-18". Section 40-1-18 has the application form which includes the terms bride and groom in appropriate sections as well as "male applicant" and "female applicant".
So the lawsuits in New Mexico are basically between Article II, section 18 "No person shall be deprived of life, liberty or property without due process of law; nor shall any person be denied equal protection of the laws. Equality of rights under law shall not be denied on account of the sex of any person. The effective date of this amendment shall be July 1, 1973. (As amended November 7, 1972, effective July 1, 1973)." and
the form in section 40-1-18 which specifically mentions "Bride/Groom" and "Male Appliant/Female Applicant".