I don't know how to tell you this, but Denmark is a sovereign state and does not derive its interpretation and objectives of copyright from the US constitution.

> Will companies now need to license "the likeness" of people too?

They already do.

https://en.m.wikipedia.org/wiki/Personality_rights

You keep saying what "we" prefer, but I wasn't aware that everyone shared the same values around privacy. Some cultures take it much more seriously than others, often in ways you wouldn't expect if you're coming from e.g. the US.