The King only cares because he has to be certain any children they have are his. He doesn't want to "waste" his harem space on a woman pregnant with someone else's child.
Or at least, that's the excuse he gives. Personally, I think the subtext is that he just considers non-virgins to be "spoiled goods", as it were.
The Hippians just care about pleasuring themselves with any woman they can.
The real oddity though is the fact that the girls having sex with Hippians causes them to be unattainable for Ares. Given his characterization, I don't think he would give a damn about virginity (hell, he'll happily marry most of the girls even after they've had sex in the harem). But if the girls are raped by anyone other than the King, they're a non-option. I suppose it's to enhance the NTR aspect of the game, but it still strikes me as odd.