Assuming both read_count.csv and all genefamilies.csv are tabulated files containing 2 columns (read_count.csv doesn't really matter), and that ´awk´ number formatting is enough for you:

for file in *.genefamilies.csv; do
 awk -v m=$(grep -F -w ${file/.genefamilies.csv} read_count.csv | awk '{print $2}') \
 'FS=OFS="\t"{NR>1 && $2*=m; print}' $file > ${file/.csv/.multiplied.csv}
done

grep extracts the multiplying factor, and the main awk loops though each line multiplying that second column for each *.genefamilies.csv existing file.

The following python approach will work, with a few caveats:

1. The use of regex to replace the number is purely because I don't know how fixed your output format is.
2. You may want to run this code on one file at a time instead of all of them, so that you can output each file to a new file (or add a write in to the code)
3. I'm assuming all your values are 0.00123. If you need to account for integers etc too, then the regex needs to change.

import sys
import os
import re

# Read the readcounts file and store a dictionary of Filename: multiplier

readcounts = {}
with open(sys.argv[1], 'r') as rc:
    next(rc)  # skip the header
    for line in rc:
        (key, val) = line.split()
        readcounts[key] = int(val)

# Read the file(s)

for file in sys.argv[2:]:
    multiplier = readcounts[file.split(os.extsep)[0]] # look up the relevant multiplier from the dict
    with open(file, 'r') as fh:
        for line in fh:
            print(re.sub(r'\d+\.\d*',
                         lambda m: str(float(m.group(0)) * multiplier),
                         line).strip("\n"))

Usage:

$ python scriptname.py readcounts.csv /path/to/files_to_multiply/*.ext

Inputs:

cat ERR260136.genefamilies.csv

 #Gene Family                                                   ERR260136_Abundance-RPKs
    UNMAPPED                                                            0.445035
    UniRef90_A0A015P9C8                                                 0.00080211
    UniRef90_A0A015P9C8|g__Bacteroides.s__Bacteroides_fragilis          0.00080211
    UniRef90_A5ZYU5                                                     0.000787149

cat ERR260140.genefamilies.csv (some sample data I made up)

#Gene Family                                                   ERR260136_Abundance-RPKs
    UNMAPPED                                                            0.1239710902
    UniRef90_A0A015P9C8                                                 0.000812739
    UniRef90_A0A015P9C8|g__Bacteroides.s__Bacteroides_fragilis          0.0031
    UniRef90_AYCHAYB                                                    0.0007871123123131

Output:

 python multiplying.py readcounts.csv *.genefamilies.csv
 #Gene Family                                                   ERR260136_Abundance-RPKs
    UNMAPPED                                                            11409246.585900001
    UniRef90_A0A015P9C8                                                 20563.485521399998
    UniRef90_A0A015P9C8|g__Bacteroides.s__Bacteroides_fragilis          20563.485521399998
    UniRef90_A5ZYU5                                                     20179.93425426
 #Gene Family                                                   ERR260136_Abundance-RPKs
    UNMAPPED                                                            2376039.3365760553
    UniRef90_A0A015P9C8                                                 15577.017442164
    UniRef90_A0A015P9C8|g__Bacteroides.s__Bacteroides_fragilis          59414.8356
    UniRef90_AYCHAYB                                                    15085.85439832861