"Going after?" You mean identifying the person? That's the "threat?" Shiver me timbers, boyo. And who ever thought a news organization would do investigative reporting, especially when someone tried to make "a funny" at their expense?
Remember, this is not "a funny" in isolation. Trump and his camp had been contemptuous towards CNN for some time now because of all of the factual reporting.
Actually the problem here with CNN is that they are holding over the guys head that CNN will expose his real name unless he alters his on-line behavior. That is the basic definition of blackmail.
Either CNN should expose his name as part of their "investigative reporting" or decide not to. Making a public statement that they will expose the guys name or not based on his future on-line behavior is completely outside the confines of proper journalism.
I expect that someone at CNN will be fired for this serious blunder.