If the exercise had given the English meaning it would have at least been possible to figure it out. I know it's possibly better to let us figure out the meaning through listening to the Chinese, but it would have helped to have pronounced the "ling" in the listening exercise. As it was, I think I translated it as June 213 - probably just as interesting a year! Though that would probably have had to have a "ling" at the beginning (0213)?
You can see it often on address plaques in China. I couldn't find an example of one with zeros but you can see how the lane (弄） number on this one is spelled out on the bottom of this sign: https://goo.gl/images/ZRgL19 If the lane number was something like 1020 it would say 一〇二〇弄. It's also commonly used to write out the date, such as on this video game: https://goo.gl/GG2v2P
The male audio is just worse in this course, poor cadence, missing sounds, weird pronunciations, in contrast to the usually very good female audio. In this case it's just plain missing the 0/零 sound. And judging by how old these complaints are, it doesn't appear they have the time, resources or inclination to fix any audio issues.
零 is the formal, "proper", way of writing it. But informally esp. when you're n a hurry "O" or "0" are used. https://en.m.wikibooks.org/wiki/Written_Chinese/Numbers
2019年6月5日 is how to write 05Jun2019 in Chinese; I have heard that 2019年6月5号 is also correct, and that 2019年6月5天 would probably be understood as well. Also, if you prefer to use Chinese characters rather than so-called "Arabic" numerals, you may. In my answer, I substituted "5th June" for the "6th June" you asked for, just so it would be clear to readers which number goes where; "06Jun2019" would be written as 2019年6月6日, etc.
THERE IS NO "〇" plus, when I was practicing this with timed practice when I was reporting it, it did not pause. Yet another discussion about a Duolingo problem that EVERYONE agrees on. I totally agree with everyone too, Andvari4, KeZhiXin1987, JHlearns, Potato132462, and I definitely agree with JesperWarn. I did a children's (kindergarten level, supposedly) Chinese workbook the other day, yesterday, and it used 零 for dates. Duo can do better.
Welcome to the 8th Century. Actually, 〇 was already used in written Chinese before the 8th Century, but gained much more popularity during and following the reign of Empress Wu (武則天 Wu3 Ze2 Tian1) during the Zhou Dynasty (16Oct690 - 22Feb705). No one was "binging on any channels" back then.