You can see at smartandsun.eu .
I only can see that switcher if I have more than one web store view in one store. As I only have one web store view in each store, the switch never appear.
I'm asking: How can I choose the store? I really need different category roots, so I really need different stores.
Yes i understand and that is what i am saying - for each store you need to assign a new store view - so you will able to see switcher on the frontend.
See here for the same - https://magento.stackexchange.com/questions/175010/store-switcher-not-showing-in-frontend-magento-2
Or there might be a case you have installed a theme so that reasons its not showing - change the theme to Luma theme and then check,
I have done the same like created a 2 main root category then assign to each store means 2 root category and 2 store and 2 store views and 1 website - and i am able to see switcher at frontend.
Hope it helps
The problem was in the theme!
Sorry for this. Problem solved!