The problem is that date takes your request quite literally and tries to use a date of 31st September (being 31st October minus one month) and then because that doesn’t exist it moves to the next day which does. The date documentation (from info date) has the following advice:
The fuzz in units can cause problems with relative items. For
example, `2003-07-31 -1 month’ might evaluate to 2003-07-01, because
2003-06-31 is an invalid date. To determine the previous month more
reliably, you can ask for the month before the 15th of the current
month. For example:$ date -R Thu, 31 Jul 2003 13:02:39 -0700 $ date --date="-1 month" +'Last month was %B?' Last month was July? $ date --date="$(date +%Y-%m-15) -1 month" +'Last month was %B!' Last month was June!