When did marriage become a basic human right?
There are many possible answers to that:
1. It was always a basic human right.
2. Around the same time freedom of association became a basic human right.
3. Around the same time the idea of basic human rights developed.
As a matter of American law, it goes back at least as far as 1967 with the unanimous Supreme Court decision in Loving v. Virginia. The UN's 1948 Universal Declaration of Human Rights also mentions marriage:
Article 16.
(1) Men and women of full age, without any limitation due to race, nationality or religion, have the right to marry and to found a family. They are entitled to equal rights as to marriage, during marriage and at its dissolution.
(2) Marriage shall be entered into only with the free and full consent of the intending spouses.
(3) The family is the natural and fundamental group unit of society and is entitled to protection by society and the State.
Why is the government involved pro or con with it to begin with?
By law and custom, marriage is a special relationship. It involves things like formalized joint property ownership, inheritance rights, power of attorney, and responsibility for and authority over children. Historically, marriage sometimes involved a legal union of two people into one person, with the woman's identity disappearing. (This is a bad thing and is no longer done in the U.S., but it was there.)
Why is it only limited to two people?
It may be possible to create a form of marriage that works for three people, but it's not necessarily straightforward. One example is automatic power of attorney when one partner is in the hospital and unable to make decisions. With a bilateral marriage, their spouse has full decision-making power. With a multilateral marriage, what do you do if two spouses disagree about treatment? I don't know if that's showstopping problem, but it doesn't exist in the context of gay marriage, which is functionally identical to straight marriage.