The .deb installation bundle for Linux should include an icon for the file association. After installing from the .deb bundle the file association itself works perfectly: you can double-click a .santa file and open Santulator. The problem is that the .santa files don't show the corresponding icon.
The icon for session files works as expected on Mac and on Windows.
Note that this is exactly the same problem as that seen in VocabHunter/VocabHunter#16. The StackOverflow question from that issue is also related.