To native German speakers: "What is his competence?" Would be a strange thing to say in English.
I think the German sentence "Was ist seine Kompetenz?" might mean "What is his expertise?" Which is OK, but it would be more normal to ask:
What is his area of expertise?
What is he qualified to do?
What is his skill?
I sometimes work with various European governments, including the German federal government, on a very specialized topic. The agencies we work with are referred to in EU legislation as "the competent authorities" because of this meaning. (The naming also leads to many jokes when we're dissatisfied.)
Officially, "further" would be correct, but most native speakers (where I live in the northern US, at least) don't distinguish between them conversationally. In English class, I was taught that you're supposed to use "farther" when there are actual physical distances involved and "further" otherwise.
I only remember it because of the movie Finding Forester; the student challenges his snooty professor: "You said my skills extend 'father' than the basketball court. 'Farther' relates to distance. 'Further' is a definition of degree. You should have said 'further'." ... The movie isn't only argumentative semantics, I promise.
Competence is a "good" English word, the fact that it is rarely used outside the Educational field, should not invalidate its meaning, it is correct, but it is certainly not as coloquial as "expertise", specially in the US, but it is used in other English speaking countries and among educated people.
The English translation is bad. I think the important thing to bear in mind is that, in addition to meaning "competence" in English, "Kompetenz" in German also means "jurisdiction" (in the sense of the geographic area or subject matter over which a court or some other agency or body has authority), "responsibility" (for an area, department, etc.) and "area of authority". Given that, I think "What is his jurisdiction?" or "What is his area of responsibility/authority?" are valid translations of this sentence (although Duo disagrees).
Here is the Collins German-English Dictionary definition for Kompetenz:
→ (area of) authority or competence; (eines Gerichts) → jurisdiction, competence; da hat er ganz eindeutig seine Kompetenzen überschritten → he has quite clearly exceeded his authority or powers here; er hat die alleinige Kompetenz, hierüber zu entscheiden → he alone has the authority or competence or is competent to decide on this issue; ich will dir nicht deine Kompetenz(en) streitig machen → I don’t want to infringe on your field; das fällt in die Kompetenz dieses Amtes → that’s the responsibility of this office; seine mangelnde Kompetenz in dieser Frage → his lack of competence in this issue (Ling) → competence
I'm really not sure what the German sentence means. The given answer does not make sense in English. Does the sentence mean "what is his responsibility"? or "authority"? or "ability"? or "skill"? or "skill level"? or "jurisdiction"? or "field of expertise"? Because each of those has a distinct meaning in English and I'm not sure if this German sentence can mean any of them, or only certain ones.
I just got marked wrong for "What is his ability?" so I wouldn't go with that..!
Duo still won't accept, "What is his authority, despite 'authority' being a definition offered.