The barbarian class doesn't really do any extra damage, until level 15+ when it gets extra attacks from Greater Mobility and Wheel of Death. The sole exception is Crimson Mist/Greater Crimson Mist, but it could easily be "wasted" in an encounter with insignificant supernatural creatures, like risen dead. All other classes may take Weapon Focus, and Improved Critical to offset Versatility. The nomad has Nomad Charge, Greater Mobility, and Mounted Mobility. The pirate has Ferocious Attack, Sneak Attack, and Greater Mobility. The soldier has Formation Combat, Greater Weapon Focus, Weapon Specialisation, and Greater Weapon Specialisation. The thief has Sneak Attack, Sneak Attack Style, and Opportunist.
The races that give +2 Str and/or the option to take Fighting-Madness have barbarian as favored class, but these races could be combined with many other classes if the player is willing to lose out on the three extra feats. A Southern Islander pirate will get these feats.
Since you're talking about incredible discrepancy in damage done, could it be that the other players have built their characters in a less than optimal manner when it comes to damage output? If the barbarian is fighting with a greatsword and the others fight with two weapons or a shield, he will generally kill his enemies significantly faster. Also, if the other players haven't maxed their Str or taken Power Attack, they will fall behind when it comes to damage.