Exactly what qualifies a race as "beastman" isn't clear, but it is mostly just a social construct used by Eorzeans to refer to peoples who worship what they seem to consider pagan gods - and most of what they consider pagan gods are mostly limited to the primals summoned in Eorzea. Notably the term isn't used in the Far East, so there's no clear divide between what qualifies a race as "beastman" vs. man. Maybe it's just used to denote those who have given up the hunter / gatherer lifestyle in favor of building up a civilization (in which case even the Xaela, who have different traditions between tribes but do have a unifying culture, qualify). Dunno.

... what I do know is that the term is unlikely to have anything to do with genetic compatibility, since the Lalafell of the First Shard (Dwarves) are considered beastmen despite having no known physiological differences.

Where do the Hrothgar sit in all this? Dunno, cuz we still don't know anything about them other than the fact they exist.