Edit section isn't ever shown alongside Observers, regardless of how their enrolment came about.
New Analytics is only ever shown alongside Students and is unrelated to how they were enrolled in the course.

Notes:

If you wish to 'remove' ('deactivate') individual students who were enrolled via the Oxford Groups tool, please follow these instructions to deactive them systematically.
We are aware of an intermittent issue, in that some teachers and course administrators (who may hold higher-level account roles), may see Remove from course alongside students who were enrolled via Oxford Groups. If the Remove option is visible to you, please do not remove such students via the People tool. Deactivate them via the Oxford Groups tool (as explained in the link above).