(There obviously is some set of axioms that is the only correct one for me, but moral realism seems to imply some set that would be the only correct one for everybody.)

Note: What I think Alicorn is saying (And I think it makes a of of sense), is that those "axioms" can be derived from the notion of "personhood" or "humanity". That is, given that humans are the way there are, from that we can derive some rules about how to behave. These rules are not truly universal, as aliens would not have them, or be in any way obliged to come up with them. (Of course, they would have there own separate system, but calling that system a form of morality would be distorting the meaning of the word.)