I have not found any rule that prevents it, but to me the spirit of the Rage ability is attacking with no time to think about self-defense, just focus on doing the most damage possible. So:
Does a barbarian/fighter benefit from the +1 from the Defense Fighting Style while Raging?
Rage only requires that the barbarian not be wearing heavy armor. It only precludes concentrating on a spell. So there's nothing in Rage to prevent applicability of the Defense fighting style. (PHB p.48, "Rage")
Defense fighting style only requires that the barbarian/fighter be wearing some armor. Per Rage, that can't be heavy armor, but light and medium armors are fine. (PHB p.72, "Defense") Note that you do lose the "unarmored defense" AC calculation from your barbarian's class features.
If you need an in-game/narrative reason, try Donan the barbarian launched into their attacks with such abandon that their opponents could hardly get a strike near them, never mind land one with any force.
Narratively, a fighting style is just that: a style. Your barbarian/fighter's style tends to make good use of their armor, even when raging. Perhaps it's a matter of accepting a hit on armor that many others would instinctively shy away from? Perhaps it's a matter of Rage-powered instincts helping them get a shield up when you'd think it's just not possible.* In any case, it's all up to you to make some narrative hay out of Rage + "better than most with armor".
* - Wulfgar, in some of R.A.Salvatore's books, is often mentioned as whipping his warhammer around faster than his opponent thought would be possible, in time to block some seemingly-easy strike. What's good for the goose....
