I'm searching for syntax / extension to include Language, Version and Formats links via shortcut similar to the one at the bottom-right of Sphinx documentation page:
Is it an reStructuredText directive or Sphinx extension that generates these links holder?
It's neither. The image states, "Free documentation hosting provided by Read the Docs", which generates this from your reStructuredText files and Sphinx documentation project.
You can read the docs about Read the Docs for versions and localization for languages. RTD builds HTML, PDF, and epub formats by default, but you can turn off PDF and epub in the admin under Advanced Settings for your project.
