Scriptural Basis for different kinds of marriage

There is no mention in the Bible either of clergy-officiated marriage ceremonies or of state-officiated marriage ceremonies. That's because neither of them existed in Judaeo-Christian society until relatively recently.

Marriage ceremonies officiated by a priest or minister emerged only in the 1500s, and took another two or three centuries after that to become accepted as the norm in all Western Christian countries. State involvement in marriage developed during the same time period in Europe. (Quick reference: Marriage -> History of Marriage, from Wikipedia. For an extensive history of marriage, see: Marriage, a History: How Love Conquered Marriage, by Stephanie Coontz.)

Since the Bible itself is silent on both clergy-officiated marriages and state-officiated marriages, which one is preferable or proper is a matter of church doctrine and interpretation, and of social and legal custom, rather than of specific Biblical teaching or narrative. So aside from opinion-based answers, which are not supported on Christianity.SE, the only way the question could be answered would be to refer to the teachings of particular branches or denominations of Christianity.

However, to provide a some biblical background, in Bible times marriage took place primarily in the context of family, social, and inter-tribal or international relationships. Most commonly marriages were arranged by the families of the bride and groom in order to form alliances, solidify inter-family ties, and benefit both parties (families, tribes, or nations) economically. Money or valuables commonly changed hands at the time a marriage was contracted, usually in the form of a bride price. The marriage became legally binding at the time it was arranged, and came fully into effect at the time it was consummated—at which time there was often a feast in celebration of the union.

The most detailed account of marriage practices in the Bible occurs in the story of the marriage of Isaac and Rebekah in Genesis 24. In line with the customs of the time, the marriage was arranged between the two families before the bride and groom even saw one another. The family of the groom gave expensive presents both to the bride herself and to the bride's family. The union was completed when Rebekah was brought back to Isaac and the marriage was consummated in the tent of Sarah, the deceased mother of the groom. (The consummation is implied rather than explicitly stated.)

Another account providing insight into ancient Hebrew marriage customs occurs in the marriage of Jacob to Leah and Rachel in Genesis 29:14–30. Since Jacob had brought no wealth with him, and was at that time cut off from his family, the arranged bride price was seven years of labor. However, there was a bit of trickery involved. Jacob had contracted to marry Rachel in return for seven years of labor. But her father Laban substituted Leah, Rachel's older sister, at the time the marriage was to be consummated. (Apparently there was plenty of good wine at the wedding feast!) Since his marriage with Leah was now consummated and could therefore not be undone according to the marriage customs of the time, Jacob was obliged to perform another seven years of labor for Laban as a bride price for Rachel.

The role of marriage in international relations can be seen in the later story of Solomon taking "700 wives, who were princesses [from the surrounding nations], and 300 concubines" (1 Kings 11:3). This built up Solomon's wealth by establishing treaty and trade relationships with the surrounding nations. But it ultimately led to the downfall of the unified kingdom of Israel under his son Rehoboam because Solomon began building shrines to the gods and goddesses of his foreign wives, contrary to the commandments of the God of Israel. For the story in its context, see 1 Kings 10–11.

The story of the wedding at Cana in John 2:1–12 offers a peek into wedding practices in New Testament times. Here there is mention of the wedding feast, which was a common practice in the ancient world. But as we would expect, there is no mention of any officiating priest, nor of any involvement of the state, since as mentioned earlier, these involvements did not develop until 1,500 years later. The Parable of the Wedding Feast in Matthew 22:1–14 and The Parable of the Ten Virgins in Matthew 25:1–12 provide additional insight into wedding practices in New Testament times and culture.

Yes, there are laws in the Bible about marriage and adultery, both in the Old Testament and in the New Testament. But none of these laws require or assume any clergy or state involvement in the wedding. Marriages were simply accepted as valid when they were formed according to the social customs of the day, as described briefly above.

About the most we can conclude from the Bible's descriptions of and laws concerning marriage, then, is that marriage is generally to be done according to the social, legal, and religious customs of the time, and that once the marriage is so formed, recognized, and accepted, the laws relating to it become binding upon the couple, and upon society in relation to them as a married couple.