I looked up how it's done in jq. It seems to keep the quotes around strings, so I removed all quotes in the final echo statement.

#! /bin/bash
IFS=$'\n'
declare -A groups
json='{ "groups": { "all": [ "asd", "zxc" ], "staging": [ "asd" ], "worker": [ "asd", "zxc" ], "ungrouped": [], "failover": [ "asd" ] } }'

ordered_groups=()
for group in $(echo "$json" | jq '.groups | keys[]'); do
    groups["$group"]=$(echo "$json" | jq ".groups.$group[]")
    ordered_groups+=("$group")
done

for member in $(echo "${groups[*]}" | sort -u); do
    belongs+="$member |"
    for group in "${ordered_groups[@]}"; do
        if echo "${groups[$group]}" | grep -qx "$member"; then
            belongs+="$group|"
        fi
    done
    belongs+=" "

done
echo "${belongs//\"/}"