Got the solution that works but is clumsy.
awk '{print $1 " " $(NF)}' FileLocal.txt > output.tmp
grep '^Read' output.tmp > output2.tmp
Output2.tmp (result):
Read #101
Read #202
Read #301
sed -i 's/Read/# Read/g' FileRemote.txt
sed -i 's/# # Read/# Read/g' FileRemote.txt
FileRemote.txt (result, all lines are tagged with "#")
Quote:
## Functions
# Read Auto Hide List #101
# Read Auto Move Win #102
# Read Yad Position #103
# Read Tiling Function #104
## Styles
# Read Fvwm Thumbnails #201
# Read Diary Thumbnails #202
# Read MyBorder Style #203
## Menus
# Read MyMenu #301
# Read Flat Menu #302
awk '
NR==FNR {a[$2] = $1; next}
$(NF) in a {$1 = a[$(NF)]}
1' output2.tmp FileRemote.txt > Result.txt
$ sed -i 's/Read Read/Read/g' Result.txt
RESULT same as my opening question.
If there is a better code, I appreciate the help. The solution is for public usage.