This link should clear things up for you: http://www.engvid.com/english-vocabulary-end-finish/
Just because he talks so confidently doesn't make him right. He talks about using finish when describing something you are doing on the one hand, but wants you to use end when talking about a relationship you are having. I can finish (bring to an end) a relationship if I want to :-) What I mean is that the guidance he gives is not entirely clear cut and there is still some wiggle room.
For me, it´s a weird to hear ¨the month finishes.¨ ¨Finish¨ to me implies a more active involvement. You can finish the dinner (finish eating), finish [doing], your homework, etc. Since a month is not animate, it cannot finish [something].
I would be interested to hear from speakers of other English varieties, though.
Its not that 'finishes' is incorrect, but at least in US English, its not nearly as common to describe a year, month, week or day as finishing as it is 'ending'. I can't speak for other English speaking countries, but since Duo Lingo uses English as spoken in the US, that could explain why it marks 'finishes' as incorrect.
Generally, you use "finish" when there is an expected outcome. The "end" comes regardless of whether anything has been accomplished or not. For example, you are at a dinner and the host decides to clear the table. The meal ended but maybe you didn't finish because you spent too much time talking.