I don't think there is a simple way to do this. You can, however, do a "one-liner" solution if you use pipes and for loops, things like that:
for file in $(ls *.tif); do sed -r 's/(.*\.[0-9]{4})([0-9]{2})(.*)/\1 \2 \3/' <<< "$file" | awk '{print "mkdir -p dstDir/Month" $2 "; cp", $1 $2 $3, "dstDir/Month" $2}' | bash ; done
Formatting this a bit:
for file in $(ls *.tif); do \
sed -r 's/(.*\.[0-9]{4})([0-9]{2})(.*)/\1 \2 \3/' <<< "$file" \
| awk '{print "mkdir -p dstDir/Month" $2 "; cp", $1 $2 $3, "dstDir/Month" $2}' \
| bash
done
This needs to be executed from the directory containing your files (see "ls *.tif). You will also need to replace "dstDir" with the name of the parent directory where "Month01" will be created.
This may not be perfect, but you can edit it, if required. Also, if you don't have bash, only zsh, replace the "bash" bit by "zsh" should still work.