"No, a different one."
That depends on the gender of what is referred to with "one". That appears to be a basic fault with the Duolingo approach of having very short sentences often making implicit assumptions about context that are not obvious.
This could be remedied by either allowing all three genders or (better) by giving more context and thus forcing the users to choose the correct gender.
It depends on the context. If that other thing is a masculine noun in the accusative case, that would work. Ex: Did you see the man from the shop? (the man is the object here) Hast du den Mann vom Laden gesehen? No, (I saw) another one. Nein, (ich habe) einen anderen (Mann) (gesehen).
In German, you may use an adjective as a noun where in English we add "one". For example, "the tall one" could be "der/die/das Große" (depending on the gender of the "one"). You could also be referring to a man and say more specifically, "the tall man" - "der große Mann".
The German word "man" (not "der Mann") refers to "one" as in "a person" (English speakers often will use the non-specific "you"). "One does not simply walk into Mordor" or "You shouldn't go into the sun without sunscreen" are instances where you would use "man".
I just have to appreciate this explanation since you used a "Lord of the Rings" quote
I've seen this before, but I don't get why the last word "one" is ommitted... e.g., "Nein, ein anderes eins."?
In German the reference to "one" is not needed, and "eins" referrs to the number 1. You can specify what other thing you're talking about with something like "Nein, ein anderes Fenster", but you wouldn't add an "ein" (or any version of ein) after "anderes".