向こう primarily means "over there" or "the opposite side (of a river, mountain, phone call, etc.)". When it's used to refer to a person, it's more like "their side". Unlike "he/she", it carries a sense of distance or opposition between the speaker and the person being referred to, and it's often used in contexts related to negotiation, conflict or mind-reading, implying the existence of two standpoints.
In most cases, someone referred to as 向こう is far away from the speaker, as in your example. But it can also be used to refer to people who are physically close when there's an emphasis on psychological distance between parties. For example, during a sports match, the opposing team is commonly referred to as 向こう. On the other hand, 相手 is used whenever two people are interacting (e.g., 会話の相手, 対戦相手, 契約相手), and there is no emphasis on the sense of distance.
If a woman consistently refers to her boyfriend as 向こう throughout a conversation, I imagine there is some psychological distance between them. They might be having an argument, or she might be wondering what he wants as a birthday gift.
向こう is not stiff or formal at all. If anything, it should be avoided in formal written documents even if they're your business rivals. In business conversations, 先方 or 相手方 is commonly used to formally refer to your business partner or rival without introducing a sense of opposition.