I don't know about the 1st story but it seems that alpha can mark omegas, and viceversa, like it happens with Emma's boss
They bite their ear not the neck to mark each other. That’s why Josh covers his right ear, Chase bit him the first time they had sex. The bite mark is meant to be permanent.
In the first story, kiss me liar, Yeonwoo (the mc) bit Keith (ml and Emma’s boss) marking him and preventing Keith from having kids with anyone other than the omega that bit him. According the kiss me liar, Omega’s can reject the mark, getting rid of it, unlike the Alphas, they can have anyone’s kid, but Alpha’s once they’re marked it’s permanent. There was a whole thing in the side story when Yeonwoo lost his memories and his bite mark disappeared. Even when Yeonwoo regained his memories, Keith still had mark him again.
If the author didn't change it: an omega mark is permanent, an alpha mark can be overwritten by another alpha. It also dissappeared when Yeonwoo suffered amnesia in the side stories of KML. But they don't make the neck, they mark the ears iirc.
And not every pair marks each other, for example Chase's parents didn't.
Think of is what if the ml bites the mcs neck in a daze of passion and didn't mind too much of it bec. he thinks he's a beta. But surprise, surprise! He's not.
Though I vaguely remember in the first story, omegas are the one who leave marks and choose their mate.