Being able to annotate chess puzzles

It would be great if one could annotate chess puzzles and make this searchable somehow. This way one could highlight typical patterns one could work on or go back to understand the full background of the puzzle.

I do not understand what you want. You can do a study with detailed analysis and link it to the forums. The puzzles on lichess are usually linked to the game with the computer analysis added to it. Can you give more detail where you want these puzzles with more annotation?