They are talking about the ex-fiance (the guy Ahjo punched). That alpha guy proposed that since Ahjo is not an omega, he can be that alpha's second partner/wife. That alpha guy basically expressed interest in Ahjo and is willing to marry him - thus elevating Ahjo's family lineage from criminals to cheabol adjacent. And since Ahjo seems stable now, his parents want Ahjo to marry that alpha guy for the sake of the family (sleeping with any omega if he needs to).

Turned him down cause you didn't want to be his "Second partner" ???? Huh. Who tf does he think mcs first partner is. Im confused