I think a relative (daughter, sister) or romantic (wife, lover) will be too predictable, and doesn't strongly explain why Vermouth's rank in the hierarchy isn't on par with Rum.
It's none of the above, I believe the Boss and Vermouth are the co-founder of the organization. They're probably the only one to have achieved anti-aging, thus they respect each other, sharing the same passion of 'living in immortality'. For some reason they decide to hide their anti-aging effect, even to the organization they establish. While the Boss stays in the shadow to hide it, Vermouth doesn't want it, she prefers to live normally.
What better way to hide her anti-aging looks than faking Sharon's death and become Chris Vineyard? to further hide it from the organization, it's logical that she won't directly be under the Boss in the hierarchy table. This is why Bourbon's claim that "she's the boss" makes sense, and that's why such information will shock everyone. If Gin, Vodka, even Rum find out that Vermouth is actually the co-founder of the organization, then it confirms the existance of anti-aging drug they've been developed all this time. It enough to make everyone in the organization rebels against her and Anokata himself.
Vermouth is the boss' favorite because she understands his passion of wanting to live forever, as simple as that.
