Is this a fallacy? If so, what is it called?

All examples of a complex whole, a class, cannot be assumed to partake of the characteristics of any subset of the whole. This is a compositional fallacy. Without entering into the alleged morality of war or capital punishment, this can be demonstrated to be fallacious because all murder is killing, but not all killing is murder. So, for example, if someone suddenly comes at you from an alley with a gun in her hand, you struggle with her, and the gun goes off, killing her, you have killed her, but you have not murdered her. There was no motive (apart from self-defense), there was no malice aforethought, there was no intent to kill (you might have, in stuggling, wrested the gun from her and prevented the killing of anyone).

Applying the standard of a compositional fallacy to your example is made difficult just because of the moral questions which arise from war and capital punishment. There is a story (perhaps apocryphal) that a man condemned to be hanged for stealing a horse from a common objected to the judge who had condemned him that it was harsh to hang a man for stealing a horse from a common. The judge is said to replied that he was not being hanged for stealing a horse from a common, but so that others would not steal horses from commons. This is the deterent argument for capital punishment--that it does not simply punish those who are executed, and remove their potential threat to society, but that it deters others from committing the same crimes. I have no comment on the value of that argument.

War is, perhaps, in some examples, a simpler matter. One can often, and i would say usually, determine who is the aggressor, and who is the victim. So, for example, when Japan bombed Hawaii without first declaring war (and even had they done so as they had planned but failed to do, there would not have been time to have warned Admiral Kimmel and General Short), the United States making war on the Empire of Japan can be held to have been justified, and the killings which resulted can be said to have been killings, but not murder. Although those Japanese officers tried for war crimes unsuccessfully attempted to present an argument that the United States had provoked the war, it was claimed by the United States that the Empire of Japan attacked without provocation, and on that basis, the killings done by the Japanese can be alleged to have been both killings and murders.

Or perhaps simply..."motives" technically cannot be ascribed to "groups". This is a confusion of psychological with sociological levels of discourse. You might call this a "fallacy of anthropomorphism".

I believe most political theorists define the State as a political system having a monopoly on the legimate use of force.
Not all forms of government are organized as states.