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This is true, I checked the second to last time I saw this factoid posted, on reddit back then.
Here are my screenshots from the ordinance from that time: https://imgur.com/a/ZkTyEYq
The rule is defined in the Animal Protection Ordinance (AniPO) link at the bottom of this page: https://www.blv.admin.ch/blv/en/home/tiere/tierschutz.html
Parmigiano Reggiano and Brie are pretty legit I would say.
I hate the term “Swiss Cheese” though. There are loads of sorts of cheese here in Switzerland. None are called just “Swiss Cheese”. If they mean Emmentaler they should just say so.