How to cluster rows in a postgresql table that match an input value or match a value from any of the other matching rows?

You can use a recursive CTE. This walks the graph and then assigns the minimum identifier in the graph for each row. Note that your data does not have a unique identifier for each row so this starts by generating one:

with recursive d as (
      select row_number() over (order by contact_id_a, contact_id_b) as id, d.*
      from duplicates d
     ),
     cte (id, contact_id_a, contact_id_b, min_id, ids, lev) as (
      select id, contact_id_a, contact_id_b, id as min_id, array[id] as ids, 1 as lev
      from d
      union all
      select d.id, d.contact_id_a, d.contact_id_b, least(d.id, cte.min_id), ids || d.id, lev + 1
      from cte join
           d
           on cte.contact_id_a = d.contact_id_a or cte.contact_id_b = d.contact_id_b
      where d.id <> ALL (cte.ids)
     )
select distinct on (id) cte.*
from cte
order by id, min_id;

The column min_id contains the grouping you want.

Here is a db<>fiddle illustrating the code.

Clustering like this is an iterative process with an unknown number of steps. I have never found a solution that can be done within a recursive query.

I have not worked on CRM in over six years, but the following function is similar to how we used to generate match groups. Doing this row-by-row did not perform well enough for our workload, and accomplishing this via host language using e.g. Java HashMap() and HashSet() and inverted indexing creates very messy code.

Assuming this schema:

\d contact_info 
                 Table "public.contact_info"
      Column      |  Type   | Collation | Nullable | Default 
------------------+---------+-----------+----------+---------
 contact_id_a     | bigint  |           |          | 
 contact_id_b     | bigint  |           |          | 
 ignore_duplicate | boolean |           |          | false
 list_id          | integer |           |          | 496

select * from contact_info ;
 contact_id_a | contact_id_b | ignore_duplicate | list_id 
--------------+--------------+------------------+---------
     16247096 |     16247097 | f                |     496
     16247096 |     16247098 | f                |     496
     16247096 |     16247099 | f                |     496
     16247097 |     16247098 | f                |     496
     16247097 |     16247099 | f                |     496
     16247098 |     16247099 | f                |     496
     16247094 |     16247095 | f                |     496
(7 rows)

This function creates two temp tables to hold intermediate clusters and then returns the result once there is no more clustering possible.

create or replace function cluster_contact() 
  returns table (clust_id bigint, contact_id bigint) 
  language plpgsql as $$
declare 
  last_count bigint := 1;
  this_count bigint := 0;
begin
  create temp table contact_match (clust_id bigint, contact_id bigint) on commit drop;
  create index cm_1 on contact_match (contact_id, clust_id);
  create index cm_2 on contact_match using hash (clust_id);
  create temp table contact_hold (clust_id bigint, contact_id bigint) on commit drop;

  with dedup as (
    select distinct least(ci.contact_id_a) as clust_id,
           greatest(ci.contact_id_b) as contact_id
      from contact_info ci
     where not ci.ignore_duplicate
  )
  insert into contact_match
    select d.clust_id, d.clust_id from dedup d
    union
    select d.clust_id, d.contact_id from dedup d;

  while last_count > this_count loop

    if this_count = 0 then 
      select count(distinct cm.clust_id) into last_count from contact_match cm;
    else 
      last_count := this_count;
    end if;

    with new_cid as (
      select cm.contact_id as clust_id_old,
             min(cm.clust_id) as clust_id_new
        from contact_match cm
       group by cm.contact_id
    )
    update contact_match
       set clust_id = nc.clust_id_new
      from new_cid nc
     where contact_match.clust_id = nc.clust_id_old;

    truncate table contact_hold;
    insert into contact_hold 
      select distinct * from contact_match;
 
    truncate table contact_match;
    insert into contact_match
      select * from contact_hold;

    select count(distinct cm.clust_id) into this_count from contact_match cm;

  end loop;

  return query select * from contact_match order by clust_id, contact_id;
end $$;

One of the biggest mental blocks I have seen developers face is neglecting to include the relationship of a contact_id to itself. This leads to disjoint handling and a mental model needlessly complicated by a left-side and a right-side.

select * from cluster_contact();
 clust_id | contact_id 
----------+------------
 16247094 |   16247094
 16247094 |   16247095
 16247096 |   16247096
 16247096 |   16247097
 16247096 |   16247098
 16247096 |   16247099
(6 rows)

Please comment if you need clarification on any of the steps in this solution or if it does not work for you.

Also, know that Levenshtein is available in fuzzystrmatch, and it works well.

If you would rather have sequential clust_id starting at 1, change your return query in the function to this:

  return query 
    select dense_rank() over (order by cm.clust_id) as clust_id, 
           cm.contact_id 
      from contact_match cm 
     order by clust_id, contact_id;

It will yield:

select * from cluster_contact();
 clust_id | contact_id 
----------+------------
        1 |   16247094
        1 |   16247095
        2 |   16247096
        2 |   16247097
        2 |   16247098
        2 |   16247099
(6 rows)