I would like to add some genres to the genre tag, where are already some genres inside. Currently, if i add a genre, the existing ones will be deleted or changed to < keep >. I'm pretty sure the program is able to do this, but me as the user isn't able to do so. Any suggestions?
It depends on how your "some genres" are currently stored.
Can you please press Alt + T in Mp3tag and show us a screenshot of one of your songs with multiple GENREs?
Ok, thank you.
One possibility would be Convert Tag -> Tag for the Field GENRE with a Format String
%GENRE%; New Genre
The semicolon (a space) and "New Genre" are freely choosable.
This takes the current content of %GENRE% and adds whatever you like.
Big thanks to you for your fast and precise help 
Just a comment that I've found various players to not handle multiple genre values in the genre field consistently. Based on advice I found from others, I use the ContentGroup (same as iTunes "Groupings") to note alternative genres/sub-genres/styles and more. That field is well supported as a searchable field, and you can be as verbose as you want with additional keywords.
Take "Santa Claus is Comin' to Town" by Bruce Springsteen...what Genre is it? Rock, Classic Rock, Pop Rock, Christmas, or Holiday? Pick one (for me all Christmas music goes into a "Christmas" genre), and add the others to the ContentGroup field.
Take "Walk This Way" by Run-D.M.C Is it? Rap, Hard Rock, or Pop?
Songs by Aretha Franklin - R&B, Soul, Blues, or Jazz?
Nirvana? Rock, Hard Rock, Alternative, or Grunge?
My personal style is to keep most songs by one artist (e.g. Aretha) in ONE main genre for that Artist (e.g. R&B), with few exceptions (e.g. Christmas music)...then use Groupings for additional metadata. Aretha singing Jazz gets listed under R&B genre, with the jazz track having Jazz in the ContentGroup.
Many many songs fit into multiple genres. My preference is to keep your genre field pretty simple. It's like in a library where a book can only be on one shelf...so it has a primary category, but the search system will help you find it under multiple keywords.
Hope this is helpful.