Create a new menu with a few sample links (not the pre-installed "main" etc)
Place the menu block for the menu in your layout and give it a unique machine name.
Place a second instance of the same menu block in your layout and give it a unique machine name different from the first.
Turn on and view theming suggestions for each with render cache on and off.
Issues & Notes:
The theme suggestions for each block should be different due to different assigned machine names but instead it duplicates the first onto the additional instances leaving no way to use a different TWIG template for each menu block.
When render cache is off we get different suggestions for each block but instead of using the assigned machine name it appends the first instance machine name with --2 etc for the additional instances.
Again note this isn't an issue with the already provided main menu. Just multiple instances of menu blocks for a new menu.