In the end, I can say with almost certainty that companion progress is tied to story progress (baring any bugs/glitches like the one for the Knight's Kira companion, which I avoided). Hope this helps.
This is my impression too, that companion conversations are tied to both affection and your progress through your class questline. It seems like there is a minor bug that you will see the 'companion wants to talk' icon when you hit the required affection level, but if you're not at the required story point you won't actually be able to talk to them. Then once you hit the right point in your story, the floodgates open and you get several conversations in a row until you hit the next stopping point for affection/story progress.