This web part differs in many points with the default language switch that ships with Kentico:
* Checks if a page on the target language exists or is published
* Uses the document URL path if existing
* Checks if a domain alias is defined for the target language
* and more...
See documentation in the web part property window
If you want to display the links in their native language, you have to make a change in your database. The value in the column CultureShortName of the table CMS_Culture needs to be translated For example, CultureCode fr-CA which has a CultureShortName French should become Français. This is not ideal, but is the only solution until Kentico will add a coloumn "CultureNativeShortName".