"Swiss cheese" is not... a "Swiss" cheese. It's just the name of a cheese, but that cheese does not come from Switzerland.
The canton of Bern makes an absolutely excellent Emmantaler. It’s the original Swiss cheese as brought to America by 19th-century Swiss immigrants to Wisconsin.
"Swiss cheese" is supposed to be similar to Emmentaler (or inspired from it?). But the fact that Swiss Emmentaler counts as Swiss cheese does not mean that Swiss cheese counts as Emmentaler, or is from Switzerland at all.