Perhaps this is a regional difference, but as a native English speaker, "He works day and night" sounds more natural when saying that he works very hard. I tend to use "night and day" only when I am trying to stress a difference, as in the sentence, "It's like night and day" (i.e. there's a big difference).
The idiom is said as "He works night and day." to mean he works every spare minute he can.
I think if someone said "He works days and nights." that I would think the person meant that he works day shifts and night shifts whatever his company wants him to do. It could mean that his schedule varies or that he does double shifts. I would ask questions to clarify that, but the expression above I would understand to not literally mean that he works 24 hours, but that he works as much as he can.
To say it literally (He works days and nights), I believe you would need "di" in Italian. At least, when you say someone "works days" (lavora di giorno) or "works nights" (lavora di notte) you need it. The sentence here is figurative, both in Italian and in English, for "He works really hard/all the time."
I understand that Duolingo is using this phrase as an expression, but is it wrong to think that "giorno" can also mean (only) "morning/mattina"? Or always include the morning and afternoon/pomeriggio (togheter) in these cases? Cause it actually means the entire day, right? Like day and night, but in this case "mattina e pomeriggio" are being used as 'giorno'.