TLDR: “War” is the inter-group version of “duel” (ie, lawful conflict). “Assassination” is the inter-group version of “murder” (ie, unlawful conflict).

My first “intuition about the intuition” is that it’s a historical consequence: During most history, things like freedom, and power and responsibility for enforcement of rules when conflicts (freedom vs. freedom) occur, were stratified. Conflicts between individuals in a family are resolved by the family (e.g. by the head thereof), conflicts between families (or individuals in different families) by tribal leaders or the kind. During feudalism the “scale” was formalized, but even before we had a large series of family → group → tribe → city → barony → kingdom → empire.

The key about this system is that attempts to “cross the borders” in this system, for instance punishing someone from a different group directly rather than invoking punishment from that group’s leadership is seen as an intrusion in that group’s affairs.

So assassination becomes seen as the between-group version of murder: going around the established rules of society. That’s something that is selected against in social environments (and has been discussed elsewhere).

By contrast, war is the “normal” result when there is no higher authority to recurse to, in a conflict of groups. Note that, analogously, for much of history duels were considered correct methods of conflict resolution between some individuals, as long as they respected some rules. So as long as, at least in theory, there are laws of war, war is considered a direct extension of that instinct. Assassination is seen as breaking rules, so it’s seen differently.

A few other points:

• war is very visible, so you can expend a lot of signaling to dehumanize the adversary.
• but assassination is supposed to be done in secret, so you can’t use propaganda as well (assassinating opposing leadership during a war is not seen as that much of a big problem; they’re all infidels/drug lords/terrorists anyway!)
• assassination was a bit harder (even now, drones are expensive), and failed assassination attempts would lead to escalation to war often, anyway
• assassination is oriented towards leaders, who have an interest to discourage, as much as they can, the concept. You can do that, e.g., via the meme that conflict is only honorable when it’s between armored knights on horses and the like. (For best results, add another meme which implies that observing that peasants are not allowed to own armor and horses is “dissent”.)