Ich bin erst. -> I am a serious [stern] man [in general] Es ist mein ernst. -> Just I am serious. "I mean it."
To me, "I am serious" can mean two completely different things: "I am serious about what I was just saying" and "I am a serious person". Of those two, "I am in earnest" means the former. From reading the rest of this discussion, I gather that the meaning of the German "es ist mein Ernst" is also the former, so it seems a reasonable translation to me.
Because the Germans say "es ist mein Ernst" in the same way that English speakers say "I am serious." You could translate it literally, "it is my earnestness," but that's just not what we say in English, even though that is what they say in German.
Just think of a German speaker reading "he is losing his marbles" and thinking that he was playing with marbles outside and some rolled under a bush and wondering why it translates to "he is going crazy. " Sometimes whole phrases have meanings that may not be equal to the sum of the meanings of the individual words.
Personally, I'm fully aware that "It is my seriousness" is highly unnatural English, and that no German person would translate "Es ist mein Ernst" into English using this literal construction, but I find that it helps me a great deal to remember the idiomatic construction in German if I use the literal translation, so I'm very glad that Duolingo accepts it. The unnaturalness of it in English helps me to remember it that much more, because there's nothing else I can associate that phrase with, which makes it easy to remember.
I would like some native speaker to weigh in on this too. "Ich bin im Ernst" was also what I put, because, like you, I have this recollection of actually hearing this before (in a movie or something may be?). A Google search yielded only a handful of examples of this phrase, which leads me to believe it is wrong. But I swear I've heard this used! Anyone?
"Es ist mein Ernst" is indeed an extremely common expression and means "I mean it seriously" or "I'm serious about this". Note that Ernst is capitalized because it is a noun. It does not describe the speaker's character. The same can also be expressed as: "Ich meine es ernst"
On the other hand, the phrase "Ich bin ernst" (ernst lower case because it is an adjective) would only describe the speaker's character. But I don't think you will ever hear it in that way. Perhaps something like: "Ich bin ernst, verantwortlich und entschlossen…" The two phrases have completely different meanings.
"Ich bin Ernst", as you wrote it, means "My name is Ernst" (because Ernst is capitalized.)
You should distinguish between "Ich bin ernst." -> some weird kind of 'I am without humour" or "My name is Ernst" [a common but old name in Germany] and "ES ist mein Ernst." -> "I mean it."
Ist das dein Ernst? Are you serious?
"Ich bin ernst" sagt niemand in Deutschland als alleinstehenden Satz wie "Ich bin klein." "Ich meine es ernst" -> You won't get any sweeties right now [as I told you before] - I mean it. Du bekommst jetzt keine Süßigkeiten. Es ist mein Ernst!