(1) Is marriage a religious or secular institution, or both? I believe it is both. Rather, I believe there are both religious and secular (civil) marriages.
(2) In so far as marriage is a religious institution, does the State have any place requiring religious organizations to perform marriage according to State mandates? I believe it does not.
The State can say that (secular) marriage licenses must be granted to any pair of consenting adults such that no such license will be granted to an individual already in possession of a valid license, regardless of gender, religion, ethnicity, sexual orientation, etc. etc. etc. However, it cannot require that your friend the clergyman perform the ceremony. Neither can it prohibit a Church from recognizing polygamous marriage, marriage between an adult and a minor, etc, though it does not have to recognize every marriage recognized by a Church.
If you want to be married in the eyes of the State, you must meet the State's requirements and no others. Meeting other requirements is fine, but not mandatory.
If you want to be married in the eyes of God, you must meet God's requirements and no others. Meeting other requirements is fine, but not mandatory.
It is perfectly valid to be married in the eyes of one but not the other.