Here's the approach I would take:

1. Loop thru files in dir2
2. tail -n +2 each file and append to same-named file in dir-1

This should be easy enough to implement, you might have to check if same-named file exists for each file you pick up in the loop, that should be easy too.

I'd much prefer if you tried coding this yourself. You won't need Perl or Python, this can be achieved using bash alone.

As a fail safe, before you begin this, chmod 400 all the files involved and print the output to stdout. Only when you see stuff working perfectly should you change the files back to the permissions they had earlier (or any convenient permission set) and use it to actually change the file content.

Extending Ram's approach:

First keep a backup of your files and try:

for fasta in `python -c 'import os; dir1=[fasta for fasta in os.listdir("`dir1`")]; dir2=[fasta for fasta in os.listdir("`dir2`") ]; print " ".join(list(set(dir1).intersection(dir2)))'`
do
tail -n +2 `dir2`/$fasta >> `dir1`/$fasta
done

dir1/dir2should named according to your directory names.

If you still want the sequence IDs to be appended, replace tail -n +2 with cat