Remove (not just unset) multiple strings from an array without knowing their positions

Seems to me that the easiest way to solve this is with nested for loops:

#!/usr/bin/env bash

a1=(cats cats.in catses dogs dogs.in dogses)
a2=(cats.in dogs.in)

for x in "${!a1[@]}"; do                # step through a1 by index
  for y in "${a2[@]}"; do               # step through a2 by content
    if [[ "${a1[x]}" = "$y" || "${a1[x]}" = "${y%.in}" ]]; then
      unset a1[x]
    fi
  done
done

declare -p a1

But depending on your actual data, the following might be better, using two separate for loops instead of nesting.

#!/usr/bin/env bash

a1=(cats cats.in catses dogs dogs.in dogses)
a2=(cats.in dogs.in)

# Flip "a2" array to "b", stripping ".in" as we go...
declare -A b=()
for x in "${!a2[@]}"; do
  b[${a2[x]%.in}]="$x"
done

# Check for the existence of the stripped version of the array content
# as an index of the associative array we created above.
for x in "${!a1[@]}"; do
  [[ -n "${b[${a1[x]%.in}]}" ]] && unset a1[$x] a1[${x%.in}]
done

declare -p a1

The advantage here would be that instead of looping through all of a2 for each item in a1, you just loop once over each array. Down sides might depend on your data. For example, if contents of a2 are very large, you might hit memory limits. Of course, I can't know that from what you included in your question; this solution works with the data you provided.

NOTE: this solution also depends on an associative array, which is a feature introduced to bash in version 4. If you're running an old version of bash, now might be a good time to upgrade. :)

A naive approach would be:

#!/bin/bash

# Checkes whether a value is in an array.
# Usage: "$value" "${array[@]}"
inarray () { 
    local n=$1 h
    shift
    for h in "$@";do
        [[ $n = "$h" ]] && return
    done
    return 1
}

a1=(cats cats.in catses dogs dogs.in dogses)
a2=(cats.in dogs.in)
result=()

for i in "${a1[@]}";do
    if ! inarray "$i" "${a2[@]}" && ! inarray "$i" "${a2[@]%.in}"; then
        result+=("$i")
    fi
done

# Checking.
printf '%s\n' "${result[@]}"

If you only want to print the values to stdout, you might instead want to use comm:

comm -23 <(printf '%s\n' "${a1[@]}"|sort -u) <(printf '%s\n' "${a2[@]%.in}" "${a2[@]}"|sort -u)