Since I have no clue how to actually work with Vue or edit rendered (?) .js files and because I’m a github beginner, I just forked nextcloud/server and tried to fiddle with javascript inside of apps/files_sharing.
After deleting the content of apps/files_sharing/src/views/CollaborationView.vue followed by git clone and make all, the template won’t be rendered and therefore I was able to remove the button “add to a project” in the sharing side menu.
Until someone gives me a hint for a proper way, I will stay with this workaround, hoping that it won’t break anything else.