@Guzzist thanks for taking the time to answer (especially as you are a Guzzi lover as well ).
The route planner on my Android app looks completely different. I can't replicate your screen.
However, I think I may have worked out how to save a shared route to my own library.
Open the link you have been given. This will open MyRouteApp app on your mobile. Click on the pencil symbol (bottom left). This will open the route editor. Now you need to make an edit! (for example: move one waypoint a millimeter). Click on the 'hamburger menu' Menu, and only then will you see the save option. Clicking on Save will automatically copy the route to your library.
I personally think this is not very intuitive as you may not want to make a change immediately to the route, but just copy it to your library.
But there we go, it works.