날 / 일 both mean "day" but the first is native Korean word and second is Sino-Korean (inherited from Chinese). 날 has broader use but 일 is also used for document type stuff like dates and calendars. 일 also means Sun (the sun could also be called 태양 or 해).
낮 is daylight hours, sunrise to sunset.
하루 is a 24 hour day. For example, to say "every day" you'd say 하루마다 and "day-by-day" 하루 하루.
And then there's also 오늘 which means "today."
There's also plenty of words for X days later/ago. 어제 / 그저께 yesterday, day before. 내일 / 내일 모래 tomorrow, day after. I can't remember the three or four count words...
Both google translate and deepl.com translate both the English "soup of the day", the French "soup du jour" and German "Tagessuppe" as "dagens soppa" which is the "not night" day. So it still implies a nattens soppa.
I would argue that in French "soup du jour" is the correct meaning, as in "today's soup". And it would otherwise be "soup de jour" as in "day soup", which doesn't exist.