Switzerland is not a member of the EU.
Correct you are. It’s their own tax money either way, though, isn’t it?
. . . which makes me wonder what authority they have to impose any fine on a village in Switzerland.
Switzerland does participate in the Schengen Area
Provisions in the treaties of the European Union
The legal basis for Schengen in the treaties of the European Union has been inserted in the Treaty establishing the European Community through Article 2, point 15 of the Treaty of Amsterdam. This inserted a new title named "Visas, asylum, immigration and other policies related to free movement of persons" into the treaty, currently numbered as Title IV, and comprising articles 61 to 69.The Treaty of Lisbon substantially amends the provisions of the articles in the title, renames the title to "Area of freedom, security and justice" and divides it into five chapters, called "General provisions", "Policies on border checks, asylum and immigration", "Judicial cooperation in civil matters", "Judicial cooperation in criminal matters", and "Police cooperation".