I’m thinking that it’s due to piloerection because the response is frequently referred to as having “chills” sent down your spine. The reason you’re calling it “chills” is not because you feel cold, it’s because it creates the same sensation as the one you might have when cold.
Piloerection stands hair follicles up on their ends, thereby increasing the amount of skin friction drag. The resulting thermal boundary layer reduces convective heat transfer and traps warm air close to the skin, producing a slight warming effect. Perhaps not very effective in modern humans, but probably useful for our ancestors.
I think some people do experience piloerector reflex during patriotic moments. I’ve heard people say “it made the hair on the back of my neck stand up” during such events. Maybe patriotism is just your body’s way of gearing itself up for a fight.