Yeah well nothing can really be uber strict in a relatively realistic world such as Middle Earth. There are always variants. Some Men probably used elven weapons once in a while when they could get their hands on them, perhaps there has been dwarves using mankind, elven, or even hobbit weapons one day
and so it's quite possible that elves used dwarven axes too. Same with real word I have no doubt that back in the middle age, some english generals or even soldiers took french or celtic weapons instead of their own.