How to add attributes to menu and menu items on the block level in THEME_preprocess_block()? With THEME_menu_tree() we can change the markup of the &lt;ul&gt; element and with THEME_menu_link() we can add css classes to the items of it easely. But what if we have an IF IS IN THIS BLOCK condition to incept?

Not THEME_menu_tree nor THEME_menu_link() know in which block the menues are, they alter, because both will alter all menus known by system. 

Two ways came in mind, but both were dead ends: 1) pass a variable from THEME_preprocess_block() to THEME_menu_tree() and THEME_menu_link() -> No, makes no sense. Still, they don't know what to do with it nor can they follow the foreach() of THEME_preprocess_block(), because they have no conditions for blocks. 2) Finding a higher entry level to pass the condition over and alter the menu attributes and its item attributes from there.

Well. Good point. But where would that be? Printing out the $variables inside THEME_preprocess_block() was like counting drops in the ocean. Can someone please enlighten me if I am right with the second thought and with where to look at?
There are actually functions available for specific menus too:


  Targeted overrides (suggestions) available for theme_menu_link() and
  theme_menu_tree()
  
  In addition to other changes to menu rendering, a Drupal 7
  theme can implement a THEMENAME_menu_tree__MENU_NAME() and/or
  THEMENAME_menu_link__MENU_NAME() function to override
  theme_menu_tree()/theme_menu_link() for a specific menu. For example,
  THEMENAME_menu_link__management() would override theme_menu_link() for
  links within the "Management" menu. This is similar to how
  "node--article.tpl.php" overrides "node.tpl.php".


Source: https://drupal.org/node/254940#theme-suggestions-for-menus