No, and I honestly don't know if BW thought out the military roles so well.
In the first game, the alliance is clearly a Starfleet-esque organization (albeit with substantially more militarization). In ME2, it's portrayed more as a political entity, and in ME3, it's a purely military organization.
I think it's all of the above, and always has been.
As I understand it, the Alliance was formed solely to co-ordinate the exploration, colonization, and development of other worlds by Earth-based jurisdictions. As such, it represents humanity's interests diplomatically, politically, and militarily to other worlds.
The entire Alliance Parliament & Prime Minister were on Arcturus Station when it was destroyed as one of the Reaper's first targets when they invaded Earth.
Just as individual countries have ambassadors, politicians, and military organizations, the whole of humanity has all of that in the Alliance. I guess I don't see the disconnect.