Did you try sudo chmod +x Ambermoon.net
? It sounds like Linux doesn’t recognize the executable as such. And no, you don’t need an Amiga emulator.
for some reason the file didn't want to start by double clicking for me either (also Mint), but I opened a terminal in the folder and wrote ./Ambermoon.net to start it that way and it worked :)
Edit: I now read you already tried the command line, so maybe this was a different situation (or you forgot the ./ for executing it)