On average, active same-gender attractive males tend to have many, many more partners than do any straight males. Some have hundreds or thousands, and precautions are not taken in many cases. This causes huge public health problems, including the spread of disease (including most HIV cases in the US) and the use of illegal drugs that help maintain this lifestyle.
Society thus has an interest in discouraging toeiva, including by not recognizing their unions as marriage. Teoiva marriage won’t really encourage exclusivity because there’s already a tradition that toeiva marriages aren’t exclusive. Its main effect will be to legitimize toeiva and encourage everyone with any such desires to take part.
Mike Huckabee’s supreme court brief mentions the public health argument, certainly a secular one, though he doesn’t talk about the details.
This secular argument is probably one of the main reasons why until the year 2000, virtually no society on earth ever recognized toeiva marriage. Men and women are complementary in this respect. When you have same-gender relationships, many of the men go nuts and wallow in sensuality with extreme excess, harming themselves, others and society. Just as in Sodom!