Generally you use some kind of tool to manage/update the mods and set them to load in the right order. While those tools may also work under Linux with Proton/Wine/etc, each app you launch typically has its own isolated folders. So in order to get it to work, you'd need to change where that mod manager app uses to use the folders that Proton/etc configured for the actual game like Skyrim. That's compared to just installing the mod configurator/launcher app and having it start Skyrim for you on Windows.

The fact that there's a 60 page guide on how to do it tells you it's not as easy as on Windows: https://www.nexusmods.com/skyrimspecialedition/mods/91500?tab=description