October is a month, so 'October' and 'the month of October' are interchangeable. (Compare '2020' and 'the year 2020'.)
I don't know much about MLB, but I know that postseason baseball takes place over a period of time (?weeks, a month), so in this case there is no real difference between 'in' and 'during' (assuming that the postseason falls entirely in October).
If there is any real difference in other cases, it would be that 'during October' implies a longer period of time (it's related to 'duration'). Compare 'Our two-day seminar takes place during October' (questionable usage), 'Our one-week conference takes place during October' (acceptable) and 'Postseason baseball takes place during October' (certainly acceptable).
Contrast a birthday, which is one day. I would have to say 'my birthday is in October' not 'my birthday is during October'.