The AUMF was a declaration of war. The Constitution doesn't lay out the specific form of a "declaration of war"; the Korean War, for instance, was fought under the aegis of a UN declaration. Wars fought under informal authorization by Congress date all the way back to Jefferson and the First Barbary War.
It's a popular myth that we're fighting an "undeclared" war against al Qaeda, but unfortunately, that's all it is. We really did declare war against al Qaeda, by name, in 2001, and because al Qaeda is a brand name and not a nation state, at war we will remain until Congress repeals the AUMF.
The comparison to WW2 is simple: during WW2, scores of German-Americans traveled to Europe to fight on behalf of the Nazis. We didn't read them their rights before we killed them. American citizenship did not shield them from the consequences of armed conflict. If al-Awlaki wanted to prosecute his cause using the mechanisms of due process, it was a decidedly stupid idea for him to take up residence in territory controlled by AQAP, where the only reasonable mechanism the US had available to it to stop him from killing innocent civilians was an airstrike.
Also, you're wrong about the President's authorization to kill whomever he labels a "terrorist". The President does not have a blanket authorization to use the military against "terrorists". He has authorization only to strike against al Qaeda and those persons and organizations who assisted in the attacks on 9/11. The President cannot authorize airstrikes against, say, Basque separatists, or the Tamil Tigers.