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The german word for coat of arms is „Wappen“. Etymologically that word is descended from „Waffen“, although the difference happened in the high middle ages.

That's because originally they were arms - it's how combatants would identify who was who on the battlefield based on what was painted on their shield and helmet.

“Arms” has the same two meanings as in German (see e.g. the College of Arms — hint: it’s not a place where they teach weaponry). “Coat” in this context refers to the actual coat worn by heralds representing the sovereign. The modern word for that outfit is “tabard” I believe.