Rather than 'errand', we would probably use 'appointment' (in Australia), i.e. "I have an appointment at the doctor" or "I have an appointment with the doctor". I've never heard errand used in such a context.
That's the version I know, and since an internet search does not yield any results for the usage of "errand" in this context, I'm starting to suspect that it's a regional variety that has found its way into the course.
Duo is an American app and website. It uses American English to keep it consistent. There is not enough room for them to accept every answer from every form of English there is. (Australian, Canadian, Irish, Indian, etc etc.)
Not suggesting it should. But it's always interesting to compare notes with other users.
I have lived in the US all my life, and I have never heard anyone say that they have an errand at the doctor!! We would say, "I have a doctor's appointment" or "I have an appointment with the doctor". If for some strange reason there would be another point in going there, and I can't imagine what it would be (maybe to finish the magazine article that you started in the waiting room?!!), we would never call that an errand. We might say, "I need to stop at (or run by) the doctor's office to (or for)..."
I mean, I'm not a native English speaker, but I feel as if I've heared more of "at the doctor's" in the movies and on TV. :)
Sorry if I confused you. You are 100% right - you will have heard that phrase. And you might also hear phrases like "I'm going to my friend's". In a way, it's shorthand for "I'm going to my friend's house". So there are words that we don't actually say/write, but we just assume that the listener/reader understands them from the context. Hopefully I haven't made that more confusing!
And hopefully Simon's clarification on the Norwegian helps more than I did.
I wonder if that might require something like "Jeg har et ærend hos legens" (i.e. with the "s" right at the end to denote a possessive); as that is really that you're at the doctor's rooms / doctor's surgery / etc. in my mind. As I'm not a native Norwegian speaker, I can't be sure on that bit, but I think I'm right on the English bit.
No corresponding genitive form in Norwegian, we just use "hos legen". "Hos" is a preposition grammaticalised from the original word for "house", so it serves the same purpose :)