The US Constitution has a Bill of Rights, which applies to everyone in the sense that everyone has those rights, but which only applies to the government in the sense that the government is the one the Constitution says can't violate those rights. For example, the 1st Amendment says the government can't infringe freedom of speech. It does not say that everyone is guaranteed the same platform to speak from, nor does it require particular private entities to provide a speaking platform to anyone who asks for it.

Saying "people have to be treated equally in equal cases" sounds like the same kind of thing: if the US Constitution had such a provision, I would expect it to work the same as the 1st Amendment, i.e., requiring the government to treat people equally in equal cases, not requiring private entities to do so. But the Dutch Constitution might work differently, which is why I asked.