You're right to think it's strange, because it is. It's a tax that was created long ago to try and close a loophole used by executives, but in today's world, it functions as a barrier to participation in the wealth created by young companies.

One could view the AMT as explicitly serving as a way to prevent poor/middle class people from becoming wealthy by way of stock options.

You get taxed on sale as well (as I understand it). I think the issue is that you are purchasing something at a price considered below market value so your estimated net worth changes.

Edit: FTR I think just taxing on sale for the entire realized gain would make sense to me as well...but I imagine this policy is to try and prevent high income individuals from dodging income tax by taking all their compensation in stock.