The state is Saxony and Saxon Switzerland is a national park within Saxony. It's a mountainous region that continues across the Czech border, where it is called Bohemian Switzerland.
I guess the logic behind the name is not much deeper than "it looks like Switzerland".
The German translation of "Switzerland" is "Schweiz", which is a word like "meadow". It just describes a particular region, which often happens to be mountainous and with lots of forests, like the actual nation also happens to be.
Maybe I'm just a silly American, but what kind of weirdo reports foreigners to police for being suspected migrants? Best case scenario ends like this where you end up looking like a racist dick. Like, I understand reporting actual crime to police, but the likelihood that you've run across nefarious immigrants out to cause harm is really low compared to the likelihood that they're legal citizens or tourists. And even in the off chance that they are illegal migrants, existing shouldn't be a crime, it's none of your business, shut your mouth and leave them alone.