Is there a solution to transfer .md files available in the Code tab of my repository directly to the wiki tab?
I would like to have one only source (the Code tab) to manage the .md files and be able to use them in different output contexts (the Wiki tab of the repository but also in an internet website).
No, I don't believe that is possible. You can clone the Wiki by doing a
git clone https://github.com/user/repo.wiki.git
and then place all .md files easily in there but I believe you can't just do that automatically.
You can, however, always put up a GitHub Pages (https://pages.github.com/) site to show off your documentation, kind of like the Wiki.