what makes a thinktank fake? no real authors? the about page seems rather standard to me - https://hanoverinstitute.com/hanover/about Every thinktank has always been established to push some viewpoint.
But in this case the "fake" label means that the think tank itself doesn't actually exist. The article expands on the topic with "apparently does not exist as a legal entity in any jurisdiction, has no physical address, and carries no named staff and no bylines on any of its reports".
So if they hired some people to set up the HIPP and posted the same material it'd be a real think tank, but because the institute doesn't seem to exist at all it is a fake think tank. That isn't a comment on the writing produced.