What is confirmed about unlocking MS is that it requires a moment of extreme trauma/ distress. Kind of like how Goku unlocked SS1, he felt extreme distress and about loosing friends and had a unfathomable need to do something about it. Some people think that MS is accessible through training but we don't know that for sure. Kinda like how Vegeta could not attain SS through training, instead it required him to feel distraught at an extreme level, temporarily giving up on his selfish needs.
In the Naruto world, MS requires something similar, extreme emotion. The easiest way to obtain this is to kill one's best friend. But other means include suddenly loosing a close friend or sibling in other ways. In Sasuke's case, he unlocked it through learning about the truth of his brother. This proves that any form of emotion can qualify to unlock MS, as long as it is dire enough.
The only real explanation is that Kakashi had a moment where he felt an extreme level of emotion, possibly through reflecting on his past failures. Remember, that in part 1, Kakashi had already lost everyone important to him. It is unconfirmed how Rin died, but it seems obvious to me that she died pre-part 1. So there is no real way he could have obtained it by normal means. There must have been a day in particular that he felt an increased amount of grief. Thats the only logical explanation.