Swift only supports a small number of linux systems. which is at most half-truth cause this is just official installers and most likely you'll use some sort of rustup (rust wasn't working on every system few years ago if you use official installer not some sort of rustenv installer or building from source etc.) you can use https://github.com/kylef/swiftenv you can use community packages https://wiki.archlinux.org/title/Swift etc.

"and so on is that languages other that swift aim to support linux in general." -> again not true linux distro dosen't change swift usage it is just official build is run for few most popular distros and you can use prebuild swift-bin on any linux repo. (arch, debina, ubuntu, centos etc. etc.) You can say the same stuff about rust/nim/go every other language that didn't have official release for some niche linux distro.

I don't think that, say, rust, offers any explicit support for any particular linux distro?

Swift here are basically saying they'll endeavour to make sure it works on those particular distros, but in general you're probably going to install Swift on Linux the same way you install rust; via your distro's package manager, and supported by your distro, not the Swift or Rust project.