Belarius

Does "one exactly" mean that a monogamous person would prioritize having a monogamous romantic relationship strongly over being single?

I don't think that's the meaning. Technically the word up until a few years ago mostly referred to the quality of a relationship itself, and less so to describe an individual's orientation. But to the extent it does now describe a singular person's orientation, I think it just means that at least when a person has a romantic relationship, they want that relationship to be their only romantic relationship, and they probably want it to be their partner's only relationship as well.

I imagine aromantic people wouldn't be described as monogamous if that's what you're getting at, but then again I think aromantic people have other relationships that could be monogamous as well (like a queer platonic situation).