If I filled the Inquisitor's boots, well...
Honestly, I'd probably be overwhelmed and intimidated by the likes of Cassandra, Leliana and Cullen at first. I like to think that I might get on with them (and my other companions) to some degree eventually, but it would take a great deal of time to get there.
Varric would be the first companion I'd gel with; we both love telling and writing stories. Plus, he's fairly easy-going; I imagine spending evenings by the fire, listening to him tell stories. I can see him being a dear friend by the end of it all.
I'd get on with Cole, too. I'd be unnerved, at first -- even taken aback -- that he is literally Compassion manifested. I can see myself growing to care for him, along with Varric -- we could be an odd little trio sitting by the hearth late at night trading tales.
Josephine's nice, though I know that "nice" doesn't automatically mean instant friendship. I do, however, see us having much in common: we are both diplomatic (at least, I'd like to think so), and obsessed with organizing and trying to execute things perfectly; I would enjoy learning how she plays the Game. And if I get on with Josephine, I'd probably get on with Leliana, too; we both adore shoes.
I like Iron Bull, but he would think me too prim. If he invited me for a drink after slaying a dragon, I'd probably refuse, tbh. Though I'd sit with him, and have a chat with his chargers. I'd probably wind up being friends with Blackwall first.
Solas and Vivienne would be the two companions who would make me feel like country bumpkins tbh. I picture myself getting along...tolerably...with either of them for a time, and then suddenly slipping up and do something that greatly offends either party, lmao. While our views on freedom of thought and will would align, I would probably make choices that would ****** Solas off (like keeping the Grey Wardens, for instance).
I'd have a contentious relationship with Viv. She would be the one I'd intensely dislike, but who I also secretly want to emulate tbh. I admire her a lot.
I want to say I would hit it off with Dorian immediately, but would initially consider our being friends highly unlikely. I suspect he's the person who I'd think I wouldn't have anything in common with, but it'll be him who winds up being one of my dearest friends by the end.
Sera could go either way, really. I like that she cares for the little people, and I'd love to have friendly roof chats where we're just pissing about, but I'd get easily annoyed with her shooting other perspectives down.
Cullen and Cassandra I'd respect, but I wouldn't be particularly close with them.
In-game, though, I make a point of being friends with Cassandra and Dorian, rain or shine, thick and thin, and regardless of who my character is. I love them too much!
Love as in: I can live with getting disapproval from my other companions, and making them leave, and breaking up with my LIs and all that jazz, but ending a game without Cassandra and Dorian as dearest friends would, frankly, be unthinkable and blasphemous.