Unlike the term “gay marriage,” the term “marriage equality” is a broader umbrella term that is inclusive of marriages between a man and a woman, two women, or two men - in short, between two people. Users of this term, however, are almost always supporters of marriage for same-sex couples who believe in the equality of relationships between same-sex couples and opposite-sex couples - and in particular, an equal distribution of rights and benefits to all married couples, regardless of their sexual orientation or gender identity.
By implication, those not supporting “marriage equality” represent “inequality” in marriage -specifically espousing the superiority of relationships between people of those of the opposite sex over those between those of the same sex. Religious conservatives thus push back against this term as making an intrinsic accusation of their own position as inherently prejudiced. From their perspective, the value placed on man-woman marriage does not necessarily mean any individual or organization is hateful, bigoted or against basic civil rights; nor is marriage between same-sex partners considered by them an essential, constitutionally-protected right.
On a more humorous note, some straight people who are not paying too much attention to the debate over marriage for same-sex couples have mistaken the term “marriage equality” to mean equality in chore distribution between a husband and a wife. It’s been used like this: “We are working on marriage equality. I take out the trash, do the dishes and mow the lawn, and she does the cooking and the laundry.”