I think your conclusion is correct, but the child comments mentioning fair use would not apply because fair use is a copyright concept, not a trademark concept. And I'm really only familiar with US laws, so I'm not even sure if fair use is a concept in Denmark or not. They have a different notion of copyright than we do in the US.
That said, I think Denmark did the wrong thing here. I think face and identity is much closer to a trademark than it is to something that is a created piece of artwork. Trademark law is a little bit narrower because it allows you to use the trademark to refer to the company, but it prohibits you from using the trademark to confuse customers about whether this is an authentic product. This feels quite analogous to the issue that Denmark is trying to address with the deep fakes, so I'm a little bit surprised on the copyright angle, since I can imagine a lot of legitimate uses for taking a picture of someone without needing their permission and distributing it. CCTVs, traffic cams, mug shots, police body cams, and the ever-increasing trend of recording in any situation that becomes tense or dangerous. Will a civilian be told by the cop that he's violating copyright because they want to film the interaction with the police? This reminds me of when cops would play Taylor Swift loudly in the background so that people couldn't post videos of the interaction on YouTube.
Let's say, as an example, a married politician having an affair with someone. Generally, news sites will publish photos with the face of the politician visible but would blur the other person. The former is clearly a person of public interest, the latter is not. Even if it's a photo taken in a public space.
The issue of money gatekeeping legal rights is another issue entirely which should be addressed for everything, not just this specific problem. It's also, in my opinion, a lot more prevalent in the US than the rest of the first world countries.
So it’s not a photo of a politician doing something bad. It’s an AI recreation of what they are alleged to have done.
The law does have to be written very carefully.
Then, when someone uses their face to promote something, someone else can repeat the face with what it promotes.
So I think the whole thing actually works in this particular case.
What specific behaviors does this forbid that weren't already forbidden?
But me (not really) on my website (I don't have one) where I trash politicians (I don't) and post a photo of said politician eating poop, that should be 'frowned upon'. (Or worse to shame an ex-gf or a colleague that 'won't yield to my sexual advances').
While reading the article though, I thought of the cases where a paparazzo takes a photo of CelebrityA, then the CelebrityA posts said photo to her Insta (without getting permission from the agency) and the agency sues her. Now (in Denmark) the CelebrityA can sue the paparazzo for taking her photo in the first place (right?). This would protect people from getting uncomfortable photos.
What we'll probably see is, celebrity look-alikes will be contacted to license out their own "features".
Andy Warhol drew images of Campbell's soup cans in paint. https://en.wikipedia.org/wiki/Campbell%27s_Soup_Cans
He controlled the copyright to that painting. That's transformative, and the result does not meaningfully affect Campbell's ability to trade.
Quoting that Wikipedia link: "Although Campbell's never pursued litigation against Warhol for his art, United States Supreme Court justices have stated that it is likely Warhol would have prevailed.", with two quotes from two Supreme Court cases.
The second such quote is from Neil Gorsuch: "Campbell's Soup seems to me an easy case because the purpose of the use for Andy Warhol was not to sell tomato soup in the supermarket...It was to induce a reaction from a viewer in a museum or in other settings."
On the other hand, were Warhol to stick his copyrighted images on a new brand of soup, that would violate trademark law as it would confuse buyers.