My Dutch relatives say "Dag" as goodbye sometimes, However "Doei" is more common. As far as I understood from learning in dolingo, the difference is the formality: "Tot ziens" is the most formal, then "Dag" which is only a little more formal then "Doei". Maybe native speakers can help us here:)
I'm not sure 'tot ziens' is the most formal. It translates to 'see you', although literally it means 'until seeing', shorthand for 'until we see eachother again'. 'Dag' is a formal way of saying goodbye, whereas 'doei' is a (very) informal way of saying goodbye. Young people use 'later' (same word in Eng/NL) too to say goodbye. And yes, I am a native speaker. :)
native dutch here.
i think the most formal way to say bye in dutch is: ik wens u nog een goede/prettige dag. litteraly translated to english you say: i wish a good/nice day to you. but this is not used very much in the netherlands. most used but less formal is: tot ziens.english translation: goodbye.
"Tot ziens" is the wish, when parting, to see someone again - It might be that the wish in not sincere, though, but is merely intended to be polite.
"Dag", is wishing that someone has a good day, which isn't necessarily limited to parting; it could also be used when meeting someone. It doesn't imply meeting again, or not meeting again.
"Doei" is a way to alienate people. The editors of the course apparently see that differently, but apart from answering the exercises I would hesitate to use it. Apart from the informality, it also has the distinction of having no historical meaning. Those that do use it, use it in parting.
When parting with someone, you could use each of them. Of these three only "Dag" can also be used in meeting. (The pronunciation differs slightly in that case, but take care as the audio doesn't handle that.) It's also the only one of these that allows a meaning of not seeing someone again.
It probably depends on the dialect or sociolect, as I would say that "Doei" is so informal that it shouldn't even be taught in this course: If your Dutch friends don't teach it to you, then you're not a close enough friend to use it.
"Ik wens u een goeden dag" is not in itself formal or informal. (It does show that it's not limited to either end of an encounter. ) Similarly, other ways to part aren't by definition formal or informal either. Mostly, the shortened versions are considered less formal than the longer ones, but other than that they are fairly similar.
It's not an error. As to whether your friend is uneducated, maybe he is, maybe he's not. Dag, in the sense of saying goodbye, is still used in the Netherlands, but probably not much by the younger generation. I would use it mostly when saying goodbye to elderly people or in more formal situations, i.e. to politicians in The Hague! :-) Saying "Dag" is definitely more respectful than the more common Doei. I guess nowadays, Dutch youth use the English "later." Remember, Duolingo is teaching you ABN or Algemeen Beschaafd Nederlands, not slang.
It's not so much an error, as a problem in the way Dutch is written. "Dag" when meeting tends to be pronounced shorter than the same when parting, but that details is usually lost in writing. Thus the "Dag" in meeting is indeed not used for Goodbye, but the same word is in fact used when parting, and it's unlikely that a Dutch friend has never heard it used so.