This is a bit of an unfortunate example because it mixes copyright law and trademark law, which are very different in their character.

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.