I guess since Klingons are a warrior culture (or at least all of the society seems to serve the needs of the warrior class/caste), then fighting against Klingons is no different to fighting against aliens.
I think Klingon society is more complex than is portrayed though. not everybody can be a warrior, somebody must make Birds of Prey, or make the warrior's armour, harvest the gagh, write operas, write books and poetry, etc.
it's probably that the warrior caste/class rule the society, and they dictate how it is governed. and the aristocracy like Worf and his ancestors have a permanent place in the warrior caste. People such as Martok had to earn a place in it.