You have to use array, not associative array, as argument to zenity command:
dialogarray=()
for idx in "${!matched_bookmarks[@]}";do
    dialogarray+=("$idx" "${matched_bookmarks[$idx]}")
done
zenity --list --title="Multi matches found" --text="Choose bookmark" \
     --column="Name" --column="URL" "${dialogarray[@]}"
Here is my version of your script:
#!/bin/bash
get_final_bookmarks_choice(){
    local -n Assoc=$1 Result=$2
    case ${#Assoc[@]}  in
        1 ) # open_bookmarks_command
            Result=${!Assoc[*]}
            return
            ;;
        0 ) # guard clause.  If no bm found.
            msg='No bookmarks match found.'
            notify-send "$msg"
            echo "$msg"
            exit 1
            ;;
        * )
            local -a zenDialog=()
            local idx
            for idx in "${!Assoc[@]}";do
                zenDialog+=("$idx" "${Assoc[$idx]}")
            done
            read -r Result < <(
                zenity --list --title="Multi matches found" \
                       --text="Choose bookmark" \
                       --column="Name" --column="URL" "${zenDialog[@]}" )
    esac
}
declare -A matched_bookmarks
matched_bookmarks=(
    ['match 1']='http://match1.com'
    ['match 2']='http://match2.com'
    ['match 3']='http://match3.com'
)
get_final_bookmarks_choice matched_bookmarks bm_name
bm_url="${matched_bookmarks[$bm_name]}"
echo "Bookmarks URL:$bm_url"
For fun, a SE based version:
#!/bin/bash
shopt -s extglob
get_final_bookmarks_choice(){ local -n Assoc=$1 Result=$2
    case ${#Assoc[@]}  in
        1 ) # open_bookmarks_command
            Result=${!Assoc[*]}
            return  ;;
        0 ) # guard clause.  If no bm found.
            msg='No bookmarks match found.'
            notify-send "$msg"
            echo "$msg"
            return 1  ;;
        * )
            local -a zArgs=(); local idx
            for idx in "${!Assoc[@]}";do
                zArgs+=("$idx" "${Assoc[$idx]}")
                while IFS=$'\r\n' read -r idx;do
                    [ "$idx" ]&&[ -z "${idx//*title*}" ]&&break;done< <(
                    wget -qO - "${Assoc[$idx]}")
                idx=${idx//<*([^>])>}  idx=${idx//&/&} 
                zArgs+=("$idx")
            done
            read -r Result < <( zenity --list --width 860 --height 200 \
                --title="Multi matches found" --text="Choose bookmark" \
                --column="Name" --column="URL" --column="Title" "${zArgs[@]}");;
    esac
}
declare -A matched_bookmarks
matched_bookmarks=( ['Stack Overflow']='https://stackoverflow.com/'
                    ['Code Golf']='https://codegolf.stackexchange.com/'
                    ['Unix & Linux']='https://unix.stackexchange.com/'   )
get_final_bookmarks_choice matched_bookmarks bm_name &&
    echo "Bookmarks URL:${matched_bookmarks[$bm_name]}"
should produce something like:
