Typically, a couple doesn't just say "Just bake the damned cake, I want it white with chocolate frosting." They want something that helps to celebrate their relationship and upcoming marriage. If I were going to marry a man at long last because of Obergfell, I would probably want something like a rainbow arrangement and two grooms on top of the cake. Wedding photographers pose pictures to accentuate the romance between the couple. Ours certainly did. If they didn't, they wouldn't be good photographers. That is why at least some wedding services are engaged in what seems to me to be more like speech than just cooking a meal. And if they are traditional Christians and they are being asked to do this for a same-sex wedding, they are being asked to produce, and do a good job of producing, speech that celebrates something that their religion says should not be celebrated.
There is a principled basis within the religion for taking this position. It's not just a cover for hating certain people. I am not saying it can't be challenged within Christianity. I have a lot of sympathy for people of Christian conviction who find that they are unavoidably gay and become convinced that the best way they can serve God is to have the kind of commitment in a gay relationship that is required of Christians in their marriages. Other Christians are convinced that such people have reached the wrong answer. In my view, neither side in this is stupid, ignorant, insane or wicked, nor is it right to call them bigoted.
On the other hand, I don't see a principled Christian reason for opposing interracial marriage. So far as I can tell, race isn't even a biblical concept, except for Jew and Gentile, and Paul explicitly bridged that barrier. If someone convinced me that they had a principled religious reason for not providing wedding services to a racially mixed wedding, I actually think I would argue that they have the right to refuse service. Some people are going to get mad at me for this, but there can be cases where freedom of religion trumps equal rights. Convincing me that the religious objection to mixed-race marriage was motivated by religious principle and not racial prejudice would be the trick.
Freedom of religion is central to any free society. For example, democracy is going to have a lot of trouble working in a country like Iraq where the two types of Muslims are unwilling to grant religious freedom to the other branch.
The comparisons between defenders of traditional marriage and the KKK don't wash. Traditional marriage defenders may, at the end of the day, be mistaken. If people can't see the real differences between them, then I strongly suspect that they are not primarily concerned about equal rights for gay people, they are primarily concerned about bashing traditional religion and bringing it down a peg, and gays are just a tool for doing that.