I like this question. The devs do endeavor to create companions with a range of different, distinct personalities and tendencies after all- as has the spectrum of DA players.
I tend to get on best with those who have a grounded, down-to-earth, mature personality, a good sense of humor, intelligence, and general sensibility and reasonableness: Varric especially due to his humor and general perspective and outlook, but also his palpable and enduring loyalty; Sigrun for similar, but there'd also likely be the attraction thingy (for me anyway) if I could get past her depresso tendencies; Lace Harding may not have been the most intelligent of people, but she was very likable and I'd get on well with her (definitely wish there'd been a full romance option); Blackwall (despite his past) with his earthy sense of humor and also that he tends to "approve" of the same things I do; Oghren too, though I'm sure I'd tease him a lot. I'd probably get on well with Dorian for his intelligence, wit, and humor, lots of good discussions and banter to have with him, though I'm not sure if his snobbiness would be off-putting; his flirtatiousness does seems to decisively outweigh his arrogance- which helps- and I'd empathize with his relationship with his parents. I'm tempted to add Zevran because I'd like and appreciate him a lot, but I doubt I'd be able to make the leap after the initial encounter to trust him after he'd tried to kill me, retaining that unfinished business with the Crows. Wynne had her grating moments, and her repeated Makerisms wore on me, but I liked her as a person a lot, would probably have been on very good terms with her. I doubt I'd be good friends with Alistair, but he's generally a good guy and loyal- and more or less reasonable up to the Landsmeet- so I'd certainly get on with him otherwise. I liked Aveline more than Alistair, was my 2nd best bud to Varric in DA2, very loyal and approachable, and reasonable even with all the drama in her life, just a bit more "establishment" than I tend to readily befriend. I liked Denek Helmi a lot as an NPC, would love to have gotten to know him more and would certainly want him as a companion in DA4.
Others of the "down-to-earth" type seem like I'd have a connection with but it wouldn't be solid. Sera can be respectably direct and honest, and I'd share a tendency to want to shake up delusions of grandeur regarding social status, but she's prone to irrationality and hyperbole- typical traits of an ideologue- not to mention her penchant for frivolity and her pointless antagonism with the Dalish, would make tolerance an ongoing challenge. Bull is charming, as is Krem, for their own brand of directness, but the merc life always tends to make loyalty dubious, and I'm not much for bonds formed through drinking heavily together (as they're not the strongest or healthiest of bonds). Cassandra and Leliana seem to mean well, so there would be grounds for trust and respect in a working relationship, but they both tend toward the irrational and religious which would irk me, Leliana also having that "girly" obsession with fashion that irritates me. I'd share a lot of Morrigan's attitude about life and would empathize with her predicament with her mother, plus there's her intelligence and lack of airs, so too bad she's a fan of selfishness and deception. Vivienne is actually a fairly sensible person, even wise and level-headed in her analysis of any given situation, so I could see there being a lot of meaningful agreements between us, but her haughtiness and preference for the nobility would be ultimately irreconcilable, not to mention her open embrace of duplicitousness. I'd have a hard time warming to the nobility-obsessed Josephine, but she was clearly hard-working and an asset, and she was never just plain in the wrong (plus she's kinda sexy), so I don't know. Cullen I loathed in DAO and tolerated in DA2, but he did seem to have had personal growth by DAI that I liked.
There's not a lot of getting to know Solas, only getting to know his tendencies and appreciating his knowledge. Not sure how anyone ever feels genuinely close with him, much less trusts him. He's a clear ideologue- "mages/spirits are #1!"- and enjoys playing mind games, so despite greatly enjoying him as a companion while a DAI player, I think we'd probably be at odds as people. That's the same with most of the rest: fun to have as DA companions, but not going to be friends of mine. I loved Sten as a character, but, no, not a friend, even after having won his respect (like I needed it). Every DA2 companion other than Varric and Aveline were like that, more walking caricatures than people, Fenris being the biggest disappointment given that his past as a slave would otherwise be intriguing and appreciable to me. Cole was great, but there's certainly no way of being best buds with him. Loghain was amazing to get as a companion, but as a person, well... go ahead, Alistair...