The Linux head draws its window with raylib and Dear ImGui, and puts its icon in the tray through the StatusNotifierItem protocol over D-Bus, which is what KDE Plasma, Cinnamon and XFCE provide out of the box.
Failure notifications go through notify-send, so they appear wherever a notification daemon is running.
GNOME Shell has no tray of its own. Install the AppIndicator and KStatusNotifierItem Support extension; without it BuildMonitor runs with the window only and says so in the log.
The window needs an X11 or Wayland session with OpenGL. On a minimal install:
sudo apt-get install libx11-6 libxrandr2 libxi6 libxcursor1 libxinerama1 libgl1
Tokens go to the Secret Service through secret-tool (package libsecret-tools) when it is installed, which is a GNOME Keyring or KWallet prompt the first time. Without it they are written to ~/.local/share/BuildMonitor/secrets, readable by the user alone.
A desktop entry in ~/.config/autostart/buildmonitor.desktop, pointing at ~/.dotnet/tools/buildmonitor.