Perhaps you're projecting your definition of moral realism—"morality is independent of subjective opinion"—onto other people who apply the phrase "moral realist" to themselves. As the IEP entry on moral realism states:
... we succeed in knowing certain moral judgments to be true. Moral realism implies some sort of literal success theory, and so moral knowledge is implied by it. Or, moral realism entails at least the possibility of such knowledge.
Now if knowledge is justified true belief ("plus some fourth thing," as they say), the above comes out to: "A moral proposition can't be true unless it is possible to have a subjectively justified, and true, opinion about it." So there is some peculiar dependence on subjectivity and opinion-formation even in moral realism. One will say, "That's not the kind of dependence I meant!" but so then I must caution that the matter is still not entirely clear.
The popularity of some sort of moral realism is grounded in a variety of overlapping considerations: there is the question of the normativity of meaning and content in general and the sister notion of metaepistemological realism, for instance. But you also have e.g. John Rawls' account of his theory of justice as objectivity-minded (A Theory of Justice §78):
[The principles of justice] are the principles that we would want everyone (including ourselves) to follow were we to take up together the appropriate general point of view. The original position defines this perspective, and its conditions also embody those of objectivity... Thus our moral principles and convictions are objective to the extent that they have been arrived at and tested by assuming this general standpoint...
And so they are somewhat "objective" questions, "Does action X promote the greatest balance of happiness over suffering?" or, "Does action X accord with the categorical imperative?", etc. (despite the subjectivity involved on other levels, of course). Since utilitarianism, Kantianism, divine-command theory, and so on are both individually and cumulatively popular enough, so will be a sense of moral questions as (sometimes, to some degree) "objective."