Translation:The lunch is neither fish nor meat.
In Sweden, fish is typically considered meat. A menu would have entries for "meat" (meaning beef and pork), "fowl" (or more probably "chicken") and "fish" but someone claiming to be vegetarian and still eat fish would be considered a bit weird (in the sense that that's not how the word is normally used).
Also, you can find people with exactly any combination of eating restrictions to that's difficult to base word distinctions on. (Personally, I'm apescetarian.)
Many, many, many people would disagree. You'd never find fish in the meat section of a menu, for example (or at least almost never). Instead, it gets its own section. Still, many people do consider fish to be meat, all the same. This is a rather divided topic, really. =P Personally, I wouldn't class fish as meat.
I think Duo is trying to introduce us to an idiom here which exists both in German ('weder Fisch noch Fleisch') and in English ('neither fish nor fowl'). However, by relating it to a meal to begin with, it's made it a bit confusing. The idiom may have its origins in food and religion, as is discussed in great detail in the comments above, but its true meaning is '(the subject) is not easily categorized'; 'it is neither one thing nor another'; 'it falls easily into no category.'
I'm a native English speaker who once upon a time spoke German fairly well (meaning I knew the German idiom as well as the English), and yet I missed seeing the idiom in both languages. Because the subject is Mittagessen, I translated it as "the lunch is neither fish nor meat," which is a literal translation (and which was accepted by Duo.) I think the introduction of this idiom might have been more successful if it had been given in its simplest form of "Es ist weder Fisch noch Fleisch."
So no, 'Fleisch' does not mean "fowl." :-) They're just two parallel idioms which did not come down into modern speech exactly the same way.
Weder Fisch noch Fleisch ist eigentlich immer in übertragenem Sinne gemeint - also geht es nicht wirklich um essbare oder schmackhafte Tiere! Es geht um jemanden oder etwas, der/die/das in keine gängige Kategorie passt! Also "zwischen 2 Stühlen sitzt" oder so! Jemand, der sich weder der einen noch der anderen Gruppe zugeörig fühlt, ist weder Fisch noch Fleisch! - zB wer sich als "divers" empfindet, ist weder mannlich noch weiblich.