call bash script with list/array as an argument

Do it like this: save the first 2 params and shift them out of positional params, then store the remaining positional params in an array

#!/bin/bash

src=$1
tgt=$2
shift 2
files=( "$@" )

echo "manipulation"
echo " src=$src"
echo " tgt=$tgt"
for ((i=0; i < ${#files[@]}; i++)); do
    echo " file $i: ${files[i]}"
done

So

$ bash file_manipulation.sh source_dir target_dir filename1 filename2
manipulation
 src=source_dir
 tgt=target_dir
 file 0: filename1
 file 1: filename2

You could also just use the positional params like an array:

for file do
    echo file: $file
done

No, you can only pass strings to a shell script (or whatever other program, in general).

What you can do, however, is to handle some special syntax for arrays you define yourself and parse manually in the shell script. For example, you could use parentheses as delimiters for an array. You would then have to escape them so that they aren't interpreted by the shell:

bash file_manipulation.sh source_dir target_dir \( filename1 filename2 \)

cat file_manipulation.sh 
#!/bin/bash
echo -e "source:\n  $1";
echo -e "target:\n  $2";
echo "files:";
#Convert a String with spaces to array, before delete parenthesis
my_array=(`echo $3 | sed 's/[()]//g'`)
for val in ${my_array[@]}; do
  echo "  $val";
done

#I add quotes to argument
bash file_manipulation.sh source_dir target_dir "(filename1 filename2)"

you get:

source:
  source_dir
target:
  target_dir
files:
  filename1
  filename2

NOTE: it's better without parenthesis

cat file_manipulation.sh 
#!/bin/bash
my_array=( $3 )
for val in ${my_array[@]}; do
  echo "$val";
done

#I add quotes to argument, and without parenthesis
bash file_manipulation.sh source_dir target_dir "filename1 filename2"

NOTE 2: filenames can't contain whitespace.