"Universalist" and "Subjectivist" aren't opposed or conflicting terms. "Subjective" simply says that moral statements are really statements about the attitudes or opinions of people (or something else with a mind). The opposing term is "objective". "Universalist" and "relativist" are on a different dimension from subjective and objective. Universal vs. relative is about how variable or not variable morality is.

You could have a metaethical theory that morality is both objective and relative. For example, you could define morality as what the law says and it will be relative from country to country as laws differ. You could also have a subjective and universal meta-ethics. Morality judgments could be statements about the attitudes of people but all people could have the same attitudes.

I take Eliezer to hold something like the latter-- moral judgments aren't about people's attitudes simpliciter: they're about what they would be if people were perfectly rational and had perfect information (he's hardly the first among philosophers, here). It is possible that the outcome of that would be more or less universal among humans or even a larger group. Or at least it some subset of attitudes might be universal. But I could be wrong about his view: I feel like I just end up reading my view into it whenever I try to describe his.