If not, and the close process still has to be initialized, what are the "consequences" of the duplicate-tag? Is there just this big fat textbox saying "This question already has an answer here: [LinkAndTitleToOtherQuestion] [X] answers" and everything else stays the same?
Duplicate questions are essentially closed in the same way as any of the other close reasons. They simply have an extra header listing the duplicates.
You'll notice that questions that are closed as duplicates no longer have an "answer this question" box below them.
Duplicates are kept around and not removed as they are essentially now linked together, via the "related questions" list, as a source of extra answers and information.