Marriage is an old and generally religious concept that is older than Christianity. In all cultures marriages had rules, and those rules varied tremendously depending on when and where you are. The rules and obligations are so different that even in the US a person who is married in one Christian Church many not be considered to be really married by another church.
Even though it’s true that many churches don’t recognize marriages performed in other churches, thought our history, members of one church accepted marriages their church did not sanction as valid and respected them in every other way that was not religious. That never meant, for example, that a married Protestant couple would be accepted as properly married in the eyes of the Catholic Church, but the Catholic’s didn’t care about anything else the Protestant couple did, including entering into a state provided “Marriage Certificate or License” that had no religious component in it.
Marriage Certificates and Licenses have only existed in the US since the mid-19th century. Before that there were religious marriages and common law marriages. For much of our history the only people who got religious marriages were those who were deeply religious, the rest just lived together?
State issued Marriage Certificates and Licenses were invented by people who wanted to restrict marriages to those unions they approved, for example preventing black people from marrying white people. Over time that purpose disappeared and was replaced with financial obligations, benefits, and inheritance rights.
State marriages do not fulfill any religious obligation, or qualify the people who get them to any religious standing in any religious institution. Since they have never had an impact on the religious institution of marriage they have been pretty much unopposed by every major religious group. Then came Gay Marriage and the world fell apart.
Many religious groups became incensed over the thought that Gay people could get a State issued Marriage Certificate or License and strongly objected, usually on religious grounds. Their religion says Gay Sex is a sin, and they don’t want the state to allow Gay people to be given a Marriage Certificate.
I wouldn’t suggest it would be right for us to force religious institutions that believe a Gay lifestyle is a sin, to accept openly Gay members in their church, or marry them, but I sure think they have no right to object if the State sells them a Marriage Certificate. Gay couples are no more or less out of compliance with their religion than Atheists, or people from other countries whose religious practices are also not acceptable in the eyes of many churches and they never object when they get married by the state. A state issued Marriage Certificate or License doesn't compete with a religious marriage, so religious people shouldn't care if the state lets people buy them for their dogs.