I'm not sure if this is possible, because i didn't think to try it while we were playing, but it would be cool if you could chain characters to players that weren't in control of them. My party of 3 took Laezel to be our tank, but we took turns for who controlled her, so if my friends warlock was controlling her, she ended up starting in the back of the fight. It would be better if he could chain her to me, the frontline light cleric, even when I wasn't in control of her for positioning. I'll have to test if that's already a thing next time we play.
Also I agree with you that it would be cool if all party members could socially interact with the companion. It would help them feel more lifelike in multiplayer.