The 'rights' angle isn't consistent. I'd love to find out why those who push the 'equal rights' principle are not advocating for those who want to marry multiple partners? Why are you not championing the 'right' of Mormons and Muslims to have multiple wives? Are they not deserving of 'equal rights' in the same way that gay people are? If you're going to push this rights barrel you should be up front and honest about the logical conclusion about what you're asking society to sign onto.
So enough about 'rights'. Unless you're advocating for polygamous relationships you're actually full,of crap when you speak about 'equal rights'.
I don't see the inconsistency at all. This discussion is not about polygamy. Is there a push to have such relationships recognised as marriage? If so, then it would be entirely appropriate to take a rights based approach. Without having put my mind to the issue, my personal observation of polygamy was while on duty in the Northwest Frontier of Pakistan. In that context it was associated with the subjugation of women. While I have lived in the United States, I have not had any exposure to the Mormon version of polygamy and whether there are associated issues with the denial of rights. So, if the question is asked, looking through the lens of rights is an appropriate way of deciding if the issue is worth pursuing. Right now, I don't know the answer to that question.
Regarding your previous discussion on the complex and emotive issues of euthanasia and abortion, I think a rights based approach is also helpful. I actually have taken the time to study these issues in the context of utilitarian ethics. I don't see an inconsistency with the ethical approach to these issues with views presented in support of same sex marriage, but probably best left for another discussion.