Due to the cruel and intense teaching that Seto was forced to endure, it has left him with little desire to wear clothes that are exposing in nature. 



In the manga we see that Gozaburo used a collar and a riding crop for the sake of corporal punishment as persuasion through his accelerated learning programme he forced Seto through. 
Gozaburo was shown to be a cruel man multiple times so assuming that he went over board at times isn't a stretch of imagination. 

As such from this Kaiba's neck was left with scarring from the collar, hence his penchant for high collared clothing at the time of Season 0. It would also be fair to assume that he has scars littered across his upper body, possibly else where that are the results of the beatings he would receive. 
But Duelist kingdom shows him with an open shirt?

As the years went on from Gozaburo not subjecting him to the same education methods, I imagine that Kaiba would have tried to use something like Bio-Oil to reduce the appearance of his scarring and heal the skin thoroughly. At this point it may have healed enough for him to feel comfortable wearing his shirt like that. However all it would take would be someone perhaps looking too much or even pointing something innocuous out to him about his neck/clavicle area that would build that self-consciousness up once more and back into the high collared outfit he wears from Battle City onwards.

Why does he cover it up? It's a refusal of acknowledgement. He spends a great deal of his time running from his past and all the emotional baggage that comes with it - completely unable to face it as he's still overwhelmed by the learned behaviour from his abuse. To acknowledge it is to show weakness, something he cannot allow to happen.

So for Kaiba to dress down or dress in a less covering way means he's incredibly comfortable with you.