What is the background of March Comes in like a Lion meets BUMP OF CHICKEN?
It's a collaboration project between Chica Umino, the author of March Comes in like a Lion and BUMP OF CHICKEN, a Japanese music band.
The background of this project, as mentioned on both Young Animal (the publisher of the manga) and BUMP OF CHICKEN's official site,

In the beginning, Chica Umino likes BUMP OF CHICKEN, and BUMP OF CHICKEN's members like March Comes in like a Lion.
They are respecting each other, they like each other's product. Well, let's do something!
After thinking for a while, the only thing that Chica Umino can do is drawing a manga, and BUMP OF CHICKEN is writing a song. They knew they have nothing else.

Thus, BUMP OF CHICKEN wrote a single, Fighter, and Chica Umino drew a spin-off manga.
As it has been mentioned on the question, the single and the manga was released on 28 November 2014. The limited edition of the manga's volume 10 included the single's CD and the limited digital delivery from Toy's Factory (BUMP OF CHICKEN's record label) included a serial number for the digital spin-off manga.
In 2016, the song was also chosen as one of the ending songs for the anime.
Regarding the MV, there's not much info (if, at all) about the production on the internet. Someone mentioned that it's Chica Umino's drawing, but there's no info about the animation/studio behind it.

Additional sources: Japanese Wikipedia