If the strings don't contain newlines, the following should work. It sorts the indices of the array by the length, using the strings themselves as the secondary sort criterion.

#!/bin/bash
array=(
    "tiny string"
    "the longest string in the list"
    "middle string"
    "medium string"
    "also a medium string"
    "short string"
)
expected=(
    "the longest string in the list"
    "also a medium string"
    "medium string"
    "middle string"
    "short string"
    "tiny string"
)

indexes=( $(
    for i in "${!array[@]}" ; do
        printf '%s %s %s\n' $i "${#array[i]}" "${array[i]}"
    done | sort -nrk2,2 -rk3 | cut -f1 -d' '
))

for i in "${indexes[@]}" ; do
    sorted+=("${array[i]}")
done

diff <(echo "${expected[@]}") \
     <(echo "${sorted[@]}")

If the strings don't contain newlines, the following should work. It sorts the indices of the array by the length, using the strings themselves as the secondary sort criterion.

#!/bin/bash
array=(
    "tiny string"
    "the longest string in the list"
    "middle string"
    "medium string"
    "also a medium string"
    "short string"
)
expected=(
    "the longest string in the list"
    "also a medium string"
    "medium string"
    "middle string"
    "short string"
    "tiny string"
)

indexes=( $(
    for i in "${!array[@]}" ; do
        printf '%s %s %s\n' $i "${#array[i]}" "${array[i]}"
    done | sort -nrk2,2 -rk3 | cut -f1 -d' '
))

for i in "${indexes[@]}" ; do
    sorted+=("${array[i]}")
done

diff <(echo "${expected[@]}") \
     <(echo "${sorted[@]}")

Note that moving to a real programming language can greatly simplify the solution, e.g. in Perl, you can just

sort { length $b <=> length $a or $a cmp $b } @array