Not an answer

It's some questions to your question: Based on your query, assume there're only two tables t1 and t2 :
Table t1 :

id|nm|
--+--+
 1|a |
 2|b |
 3|c |

Table t2 :

id|nm|
--+--+
 1|a |
11|ab|
12|ac|
 2|b |
21|bb|
22|bc|
 3|c |
31|cb|
32|cc|
 4|d |
41|db|
42|dd|

By apply your query logic:

select t1.id  as t1_id,
       t2.id  as t2_id,
       t1.nm  as t1_nm,
       t2.nm  as t2_nm
  from t1
  join t2
    on t1.id = t2.id
    or t2.nm like t1.nm || '%';

We get:

t1_id|t2_id|t1_nm|t2_nm|
-----+-----+-----+-----+
    1|    1|a    |a    |
    1|   11|a    |ab   |
    1|   12|a    |ac   |
    2|    2|b    |b    |
    2|   21|b    |bb   |
    2|   22|b    |bc   |
    3|    3|c    |c    |
    3|   31|c    |cb   |
    3|   32|c    |cc   |

You can see that even t1.id = t2.id , it doesn't stop the comparison t2.nm like t1.nm || '%' . Thus, the inner join in the query is actually a nested loop as:

for each row r1 in t1
   for each row r2 in t2
      compare r1 and r2
      if match then return (r1, r2)
               else next

Therefore, the join is like cross join (Cartesian product of rows from the tables in the join. ie. 5k x 10k = 50m) during execution because it tries to find every match between t1.nm and t2.nm regardless if t1.id=t2.id or not.
If the business rule is: when t1.id=t2.id , then remove the row from both tables and do the comparison for the rest. You may want to modify your query into a few steps:
Step 1. Do item_code join only and get matched_items
Step 2. Derive unmatched_items_t[1..6] by subtract matched_items from table_[1..6]
Step 3. Do string match comparison between unmatched_items_t1 and unmatched_items_t[2..6] : (t1 vs t2), (t1 vs t3), (t1 vs t4), (t1 vs t5), (t1 vs t6)
Step 4. Aggregate the 5 comparison results to t1.item_code level.

UPDATE: Could be an answer

Assume there are three tables t1 , t2 , t3 as below:

create table t1 (cd int primary key, nm varchar);
create table t2 (cd int primary key, nm varchar);
create table t3 (cd int primary key, nm varchar);

insert into t1
values
(1, 'a'),
(2, 'b'),
(3, 'c'),
(4, 'd'),
(5, 'e');

insert into t2
values
( 2, 'b'),
(21, 'a 2'),
(22, 'd 2'),
(23, 'x');

insert into t3
values
( 3, 'c'),
(31, 'a 3'),
(32, 'e 3'),
(33, 'y');

We want to do code match and then name match:

• cte_universe : this is the universe of item's code and name from t1's perspective
• cte_t1_cd_matched : code-matched data by comparing t1, t2, t3
• cte_t1_cd_unmatched : code not matched in t1 from the above
• cte_[t2|t3]_cd_unmatched : code not matched in t2, t3
• cte_nm_matched_flat : name-matched by name by table (src)
• cte_nm_matched : aggregate the above to code level
• output code-matched data union all name-matched data

with cte_item_universe as (
select t1.cd,
       t1.nm,
       case when t2.cd is null then 0 else 1 end is_t2_cd_matched,
       case when t3.cd is null then 0 else 1 end is_t3_cd_matched
  from t1 
  left
  join t2 using (cd)
  left
  join t3 using (cd)),
cte_t1_cd_matched as (
select cd, nm from cte_item_universe where is_t2_cd_matched + is_t3_cd_matched > 0),
cte_t1_cd_unmatched as (
select cd, nm from cte_item_universe where is_t2_cd_matched + is_t3_cd_matched = 0),
cte_t2_cd_unmatched as (
select t.cd, t.nm from t2 t left join cte_t1_cd_matched m using (cd) where m.cd is null),
cte_t3_cd_unmatched as (
select t.cd, t.nm from t3 t left join cte_t1_cd_matched m using (cd) where m.cd is null),
cte_nm_matched_flat as (
select s.cd, t.nm, 't2' as src
  from cte_t1_cd_unmatched s, cte_t2_cd_unmatched t
 where t.nm like s.nm || '%'
union all
select s.cd, t.nm, 't3' as src
  from cte_t1_cd_unmatched s, cte_t3_cd_unmatched t
 where t.nm like s.nm || '%'),
cte_nm_matched as (
select cd, string_agg(nm || ' (' || src || ') ', ',') as nm
  from cte_nm_matched_flat
 group by cd)
select *, 'code' as reason from cte_t1_cd_matched
union all
select *, 'name' as reason from cte_nm_matched;

Output:

cd|nm                 |reason|
--+-------------------+------+
 2|b                  |code  |
 3|c                  |code  |
 1|a 2 (t2) ,a 3 (t3) |name  |
 4|d 2 (t2)           |name  |
 5|e 3 (t3)           |name  |