Arguments against miscegenation were usually based around the fall of the white race, or a toppling of the status quo. So, yes, they were similar to arguments against same-sex marriage in that both feared change would arise from the legalization of either.

As for religious reasons, I don't think there were any against miscegenation. The only example I can find of a religious argument is this one from the trial of Richard Loving and Mildred Jeter in Virginia in 1967:

This governmental interest in selective breeding led the trial judge to declare "Almighty God created the races white, black, yellow, malay and red, and he placed them on separate continents... He did not intend for them to mix."

The argument only invokes God, and does not refer to biblical theory. However, it is a key part of racial theory, and that was often informed by religious beliefs. According to biblical history, the sons of Noah were the ones that repopulated the earth, being the last ones alive. Three of those sons, Japheth, Shem, and Ham, were used to classify different ethnic groups in early racial theory. Those descended off of Japheth were in Europe, Shem's descendants in Asia, and Ham's in Africa. The Bible also makes reference to a curse by Noah on Canaan, although this was misinterpreted in Europe as "the Curse of Ham". As Africans were supposedly the "sons of Ham", they too were cursed, and seen as inferior. This was a justification of the slave trade and the inferiority of Africans in later centuries.

So while miscegenation wasn't argued against in religious concepts, it was argues against with racial ones, of which one origin was found in religion. It's important to remember that this religious rationalization of racial inferiority wasn't the only concept - phrenology, the Great Chain of Being, and other means were used to classify races as lower than the Caucasians of Europe.