I'm not sure if I get the requirement correctly. If a build is started by scheduled build plugin, the behavior is basically the same like starting a build using the "Build now" button. This means that the user who scheduled the build should be logged in the first line of the console output:
If this is not the case in your build, you are maybe able to get the user information manually by using https://wiki.jenkins.io/display/JENKINS/Build+User+Vars+Plugin
What exactly do you mean by assigning another user? Running a job as another jenkins user? Or provide user credentials as a build parameter?