I found the way how to easily make "universal" Document ID links visible for all documents in our portal thru Search Center, because users don't want to remake all links in documents everytime we change company structure. I created site calculated column for Document content type and users can add this column to any view but link inside is text only. No big deal, but and I want more. My code:
="https://portal/sites/search/Pages/results.aspx?k="&[Document ID Value]
Or is there any other way how to bring this URL to users? Thank you.
 Solved, our developer made a feature for us (and yes, it is a kind of script attach to link in context menu after all...). Thank you