In all Doom engine games, you are always able to pick up a key, even if you cheated to get the key.
In Vavoom, this is not the case. You cannot pick up the same key again for whatever the reason.
To my knowledge, only Strife has a level where you pick up two of the same key. It is the Oracle key that is dropped by the Acolytes that are near the door to the Oracle. You are unable to pick up the duplicate key in Vavoom.
Edit: Update version number