If you want to make Godot available to all users, it's best to place the binary in /usr/local/bin and rename it to something like godot.
You can then create a .desktop file to make it available in the menu, containing this:
[Desktop Entry]
Version=1.0
Type=Application
Name=Godot
GenericName=Game engine
Comment=2D and 3D game engine
Exec=/usr/local/bin/godot
Terminal=false
Categories=Game;
Place that newly created file in /usr/local/share/applications, and you should be good to go.
I haven't looked into providing an icon to the .desktop file, but if you need an icon, here's one in PNG format.
Note that for placing files in /usr/local, you need administrative rights (you can use sudo mv <source> <destination> to move files using the command line).