The store is a separate company, not based in Switzerland
It turns out that this isn't actually a subsidiary of Tile.com - despite the automatic redirection from their site - it's a separate company, which I only figured out after purchasing.
The merchant behind this (Scalefast SAS) don't have a shop in Switzerland. They sent my order from the Netherlands, without informing me when I made my purchase. I just got a bill from customs that increased the purchase price by 33% thanks to the customs service fee. If I had known, I would have ordered from one of the local shops - which is, I assume, exactly why they hide this information.
According to reviews in the Apple App Store, you also don't get replacement batteries under the premium subscription in Switzerland, which would make sense if they don't have a retailer in Switzerland. I suggest Swiss buyers check what limitations the premium subscription has before potentially wasting money here too.
Add to this the fact that you need separate logins for tile.com / the Tile app and ch.tile.com because, well, separate companies, and there really is zero reason to buy the hardware from them. It's definitely not a mistake I'll be making again.
So, I haven't even received the tiles yet but I'm regretting my purchase already. Here's hoping the actual app is better than the shopping experience.