Folder comparison

The better way

Don't parse ls; use globs instead. In fact you're already using globs, just wrapping them in ls, which is pointless. You just need nullglob turned on for when there are no matches.

Also avoiding cd simplifies things.

#!/bin/bash

shopt -s nullglob

dir1=A
dir2=B

for dir in "$dir1"/*/; do
    basename="$(basename -- "$dir")"
    dirs_match=( "$dir2/$basename"*/ )
    case ${#dirs_match[@]} in
    0)
        ;;
    1)
        echo "Unique match for $dir: ${dirs_match[*]}"
        ;;
    *)
        echo "Multiple matches for $dir: ${dirs_match[*]}" >&2
        ;;
    esac
done

Output:

Unique match for A/child-1/: B/child-1-some-text/
Unique match for A/child-2/: B/child-2-more-text/
Multiple matches for A/child-3/: B/child-3-nothing/ B/child-3-something/

I added B/child-3-something to test the secondary requirement. This creates the directory structure for testing:

mkdir -p A/child-{1..5} B/child-{1-some-text,2-more-text,3-nothing,3-something,6-random-text,7-more-random-text}

By the way, ShellCheck is very useful for finding problems in shell scripts.

Calling ls on a non existent folder throws the error message that you encountered. The easy way is to just ignore this by replacing line 5 in your script with this: new_dirs=(`ls -1d $f* 2> /dev/null`);.