Knowing this is a Manga, I'm pretty sure Rio has fully functioning male and female genitalia, as that is the case 99% of the time as per my experience consuming a shit ton of Futanari content.
As for Rio's behavior here, remembering some of her past and what she just explained here, I wouldn't be shocked if she was raised by her parents - or perhaps servants given the duty of raising her, - as a girl, and because of this, she has been forced into thinking that she needs to produce heirs to continue the family-line or someone else's as a political deal.
That because of this forced mindset, she's only ever thought of dating guys, and whenever they find out about her being a hermaphrodite and rejecting her for having a penis, the mindset of being a girl makes the rejection all the more worse.
This is my own speculation, as I don't know the timeline between the self-mutilation attempt and the apparent mutilation of the two boys, but I'd suspect that she was rejected by both boys, then in her despair and self-loathing, she attempted to cut off her breasts because she thought she could no longer be a girl. When failing to finish before falling unconscious, she eventually came to like herself for who she is, but still resented the two boys for rejecting her, so she decided to get revenge by doing to them what the attributed to them making her do to herself, mutilation.
However, even though she likes herself for who she is, she's likely still stuck on the idea that she needs to be the one to provide an heir, as in, to give birth to an heir, and she still yearns for acceptance from someone, so being desperate to achieve that goal and her wish for acceptance, she's now forcing herself onto Kurono thinking that's the only way for her to go.
I'm hoping Kurono will gently stop her, let her know that she can just be herself and not worry about any obligations or any such things, take things as they come and enjoy her life however she wishes, whether that means to continue being a girl and dating boys, or it means being a boy and dating girls for once, or a mixture of both while not caring about obligations of finding a partner to continue the family line and worry about that later once the time comes for such. That he lets her know he'll be by her side no matter what, as a friend - or perhaps more, later on, should things naturally develop into such, - if she needs him for anything.
