So, does this mean Azumarill's Attack is calculated using Base 100 Attack, or is its Attack stat calculated using base 50 Attack, then doubled?
The attack stat of your Azumarill is doubled. Not the base stat itself.
So say your Azumarill had 150 Attack stat (in its summary). When in battle, this stat would double thanks to its ability to 300 (though this change wouldn't be visible).