I think the translation of “オナシップ” as “Ownership” is wrong. Firstly, the standard Japanization of “owner” is “オーナー”, not “オナ”. Secondly, there's no obvious meaning to “Ownership”. Thirdly, the alternate explanation simply works too well, and that is: it's a combination of “ona-” [オナ-] (meaning “masturbation”)* and “-ship” [-シップ]. This makes much more sense to me, as the President and Kosakura used each other for sex and do not have an established romantic relationship.
However, I could be wrong. “owner” is also sometimes Japanized as “オーナ”, and removing the other sound extender isn't that much of a stretch. Also, if it is what was meant, “Ownership” could be referring to how Kosakura owns the President (sexually), not Miyamura, especially with the smirk at the end.
But I don't buy it. “オーナ” is already a rare variation compared to “オーナー”. “オナ” would have to be super rare. I couldn't find any documentation or examples of it. No, all I found was words relating to masturbation. And that's because…
*“オナ” is a root meaning ‘masturbation’. Japanese imported the German “Onanie” as “オナニー” and then used this root to great effect. Most widely known from “onahole” (fleshlight) probably. Additionally, “Onanie” comes from “Onan”, the 2nd son of Judah (in the Bible). In English, we really only have “onanism” from this.