A government can permit whatever contracts to be drafted they choose. In the past, governments have permitted contracts between men and boys for sexual relations in the Greco-Roman empire, contracts for slave ownership, contracts for piracy and murder as government sponsored privateers, etc... just as a few random examples of heinous contracts permitted by government. If you can think of it and it's physically possible, it's probably been done under a government contract of one form or another in some place and time.
What we're talking about is marriage within the Christian worldview.
It is no small thing that Paul uses the differentiation between men and women in their designed role as husband and wife as the portrait of the relationship between Christ and His church.
Look at Genesis 5:2, "He [God] created them male (1) and female (2), and He blessed them and named them Man (3) in the day when they were created."
1 male = זָכָ֥ר
2 female = וּנְקֵבָ֖ה
3 Man = אָדָ֔ם
1 male means a biological male not just a gender role. 2 female means a biological female not just a gender role. 3 means mankind or humanity encompassing both biological males and females with the gender roles that each were designed for.
God created males and females biologically distinct who only together receive the mandate to be fruitful and multiply and have dominion over creation (Gen 1:27–28). Together a biological male and biological woman are complete (Gen 2:18–25).
The very pairing of man and woman in the assertion "male and female he created them" (Gen 1:27 RSV) implies that together the two biologically different sexes make up humanity and the great manifestation of this essential unity is marriage (Gen 2:23–24) between a man and a woman.
In the Genesis 1 creation account, Adam emerges as "the human" (ha’a?am, Gen 1:26–27), differentiated as a biological male and biological female (e.g. "male זָכָ֥ר and female וּנְקֵבָ֖ה He [God] created them"
Genesis 1:27 Hebrew Text Analysis).
If you understand ancient Hebrew, you can clearly see the language COMPLETELY REFUTES the idea that marriage can be between homosexuals.
When Genesis 1-3 are lined up next to Ephesians 5 we see that husband and wife means a biologically differentiated man and woman and that this was God's intent from the beginning. Christ did not come to destroy gender distinctions between biologically differentiated men and women but to repair them so that they might operate in such a way that shows His glory
People who choose to engage in homosexual acts malign scripture, malign God's intent, malign God's plan for humanity, etc... to justify their homosexually immoral behavior.
The government permitting it doesn't mean God recognizes it as a valid marriage. It's an abomination done in rebellion of what God established for humanity.
You can argue that same-sex marriage is a sin - but you're wrong to say there is no such thing.
Marriage is a contract between two people before God, according to Christians. However, atheists and Hindus get marriage all the time. Sure, it might not be marriage as recognized by God, but it's still marriage.