If Kurama is really the living embodiment of hatred (not some self proclaim title), then what about the other Tailed Beasts? Are they the embodiment of something else?
As per naruto.wikia.com
Long years of receiving humanity's negative treatment caused Kurama to develop intense hostility and distrust against them, even going as far as proclaiming itself to be the living embodiment of hatred.
So, he was not embodiment of hatred by itself from start but became like this later on. So its not a tailed beast characteristic to represent as embodiment of single emotion per tailed beast.