your car is 6 years old and depending on if the battery has ever been changed in it, a battery only lasts a maximum of five years. So at this point you should be on your second battery. Also there's no telling how long an alternator will actually last but we're talkin about six years and it's about time for things to start breaking and wearing out.
I've used a lot of devices that plug into that diagnostic port that the sync up and drive plugs into. The only Power it uses is the power any lights on it. if it did not drain your battery all of those first few months there's no way it just did it overnight. It definitely did not destroy your alternator.
it's quite possible that the alternator was bad and was either discharging the battery or not charging it properly.
it's a catch-22 because if the battery is constantly too weak for the alternator to keep up the alternator wears out prematurely and if the alternator is not charging properly the battery wears out.
while it's possible that the sync up and drive can use power while plugged in the port itself should not necessarily be on when the car is off. even if the port stays on is not really using much power so that's the answer I would think a dealership would give you ...that they would have to blame it on something else.