War as practised by humans and chimps is fundementally different, it is a coordinated social activity most animals simply don't comprehend let alone practice.
Two words: "kin selection".
Humans and chimps are social primates. We live in groups that are relatively close to us, genetically, although humans practice exogamy (mating outside their immediate kin group) a lot more aggressively than any of our cousins.
So to say "fighting for mates is always one vs one" is to say "kin selection does not exist", which it manifestly does.
War is mate competition carried out by other means. There is no other rational for it (war is always economically irrational, although this is not generally understood because it "just makes sense" to so many people that war is somehow a good idea.)
No individual of any species ever under any circumstances kills another member of the same species for any reason other than mate competition, either for themselves or for close kin (this is not quite true, but it should be the starting point of any analysis of deadly interpersonal violence.) Killing has zero to do with hunting behaviour--both male and female bonobos hunt, and don't kill each other. Elk are vegetarian, and do kill each other. Only when reproduction is on the line does the risk of being killed in a potentially deadly fight make evolutionary sense, in humans as well as in other species.
In humans, war creates all kinds of mating opportunities beyond the simple-minded "conquer the enemy and rape their women" scenario. In particular, it creates opportunities on the home front of all kinds, and that is a very fundamental part of its completely irrational appeal.