I think a intermediate addition is needed to be added to the game for this suggestion to work well. Item tags should be more generalized so that shields, tools, weapons, and armor all shared the same basic type of "item" and then tags dictated how the item could be used and with what skill. This only makes a small difference with the current setup of weapons and items (attacks added to shields, block added to weapons, attacks added to gauntlets/spiked gauntlets, specific attacks added to various tools). But it would allows for the additional combinations that would make one or two unique cultural tools make sense.
Let me give a few examples of what I mean:
In it's early history, a mining site of a dwarf civ is attacked and the miners defend it with picks. Afterwards, the mining site has its weaponsmiths craft larger heavier picks that require more metal, but serve as a better weapon. The dwarf civ favors these picks as dual use weapons/tools and they become a culturally favored weapons over time.
http://en.wikipedia.org/wiki/Horseman's_pick A human wood cutter becomes an explorer in world gen. Along the way, he master swordsmanship. After becoming famous, he commissions a sword that can also be used to cut wood. The tool never becomes popular as a weapon for the civ, but peasant woodcutters favor it to defend themselves with and as a tool.
http://en.wikipedia.org/wiki/Machete Because of the civ's king's personal preference for spears, spears are the favored weapon of his civ. A skilled weaponsmith lengthens the cutting edge, trying to get a better slashing weapon, while following his culture. The weapons is only used by full time warriors that can afford its price over regular spears.
http://en.wikipedia.org/wiki/Naginata A legendary mason/engraver favors his bush hammer and brick hammer. He requests a single tool with both surfaces. It becomes a common cultural practice to make double headed hammers with specialized heads on each side.
http://en.wikipedia.org/wiki/Stonemason%27s_hammer Note that for a lot of these suggestions require more tags for weapon properties and tool uses. Tags to specify weapon reach (for example to make a naginata different from a sword). Tags for tool use (a mason might need a general use hammer, a chisel, and various special hammers).
Also, I think in general GavJ is right. Most weapons and tools will undergo a convergent evolution over time. I think only one or two special weapons/tools per culture would make sense practically and thematically. So all civs would get axes, picks, swords, shields, stoneworking tools, etc. Each civ would develop a few specialized versions with extra tools uses or attacks according to various worldgen events (historical leader preferences, tools used as weapons during uprisings, etc.).