Thanks for the replies. I want to like that balcony scene, in fact initially I did, because I always rail against 1) the lack of interest/concern/support from the NPCs towards the protagonist in Bioware games (instead of it all being a one-way street in the other direction), and 2) the lack of platonic friendship moments. So seeing this was really touching and right up my alley ... at first. Then I thought wait, this feels really out of character. Josephine's middle name is "Proper Protocol and Manners", and when I thought about it I just couldn't imagine her getting so touchy-feely, especially in public.
Then I learned that this is essentially a romance scene stand-in and got quite bitter about it -- it felt like it wasn't about friendly support at all, but about the game telling me "Why the hell aren't you lip-locked with someone yet, loser? Here's your consolation prize. Better go for it, it's your last chance." Cynical, yes, but given the how romance is usually everything and friendship is often disparaged as "just friends" or "the friendzone", the cynicism doesn't come out of the blue.
Basically, I'd like to see I get can a little less jaded about it by asking people who know Josephine better if this scene can make sense as actual friendly support, despite my impression that the behavior is out of character for her.
Don't get me wrong, I like Josie -- she's not a favourite, but she's solidly and safely in the camp of "characters I respect and appreciate". One reason for that is that she's one of the few characters who ever express the briefest and tiniest bit of interest in or concern for the protagonist -- she's the one whom you can talk to a bit about your background in Haven, and in Skyhold she asks if the mark is painful. I felt that was genuine, not just because the protagonist is her colleague/boss and she has to get along with her, but because she's a good person at heart, someone who would care. Her friendship with Leliana is a rare case of a strong and purely positive bond between two female characters, and beyond that, I'm just damn grateful that she handles all the political song-and-dance and the annoying nobles with such aplomb.
But she is all about the proper manners, and this is a public and hugely important occasion, so her "paste new dialogue over the romance-scene animations" moment just doesn't work for me -- there's nothing before or after that scene that strikes me as supporting a relationship that is close/"intimate"/relaxed enough for her to ignore protocol and touch the Inquisitor like that. I could see it with Dorian and possibly Cassandra, but essentially nobody else.
Maybe it's due to the fact that touch is hard to animate/pose so it's essentially non-existent outside of romance scenes. When it does happen and it's not hostile (Cassandra grabbing our shackled hand in the very beginning/Corypheus hoisting us up by the same hand during In Your Heart Shall Burn) I'm just shocked on principle of not being used to seeing it at all. 
Long post, a bit rambly, but I hope it makes some sense.