It'd be slightly amusing to see the Pentagon work Amazon over on negotiations like Amazon worked cities over on HQ2. Amazon ran into an entity that was larger than itself.
You think Amazon cares? They will happily eat this as a loss-leader. Neither MS or Amazon is fighting over this for a measly $1B a year. They are fighting over this for $100s of Billions of contracts with other parts of the government.
Not just other parts of government. There are a lot of big defense companies (ie Boeing, Lockheed, Raytheon, etc). Being on the same cloud as the DoD is going to be a big selling point.