Postgres 13

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

create table "name_group_links" (
  "name"       varchar(255),
  "group_id"   int
);

insert into
  name_group_links
values
--------------------------------------------------------------------------------
  ('1A',11), ('1A',12),
             ('1B',12), ('1B',13),
                        ('1C',13), ('1C',14),
                                   ('1D',14), ('1D',15),
  ('1E',11),                                             ('1E',15),
--------------------------------------------------------------------------------
  ('2A',21), ('2A',22),
             ('2B',22), ('2B',23),
                        ('2C',23), ('2C',24),
                                   ('2D',24), ('2D',25),
--------------------------------------------------------------------------------
  ('3A',31), ('3A',32), ('3A',33),
             ('3B',32),
--------------------------------------------------------------------------------
  ('4A',41), ('4A',42),
             ('4B',42), ('4B',43),            ('4D',45), ('4D',46),
                        ('4C',43), ('4C',44), ('4C',45),
                                              ('4E',45),            ('4E',47),
--------------------------------------------------------------------------------
  ('5A',51),            ('5A',53),
                        ('5B',53),            ('5B',55),
                                              ('5C',55),            ('5C',57),
                                                         ('5D',56), ('5D',57),
                                   ('5E',54),            ('5E',56),
             ('5F',52),            ('5F',54),
--------------------------------------------------------------------------------
  ('0Z',01)
--------------------------------------------------------------------------------
;

46 rows affected

WITH
  RECURSIVE "membership"
AS
(
  SELECT
    "name",
    "group_id",
    (SELECT COUNT(DISTINCT "name") FROM "name_group_links")   AS unique_names,
    0                                                         AS current_depth,
    COUNT(*) OVER ()                                          AS current_links
  FROM
    "name_group_links"

UNION ALL

SELECT
    *,
    COUNT(*) OVER ()   AS "current_links"
  FROM
  (
    SELECT DISTINCT
      "name",
      MIN("min_group_by_name") OVER (PARTITION BY "group_id")   AS "group",
      "unique_names",
      "current_depth" + 1                                       AS "current_depth"
    FROM
    (
      SELECT
        *,
        MIN("group_id") OVER (PARTITION BY "name")  AS "min_group_by_name"
      FROM
        "membership"
      WHERE
        "current_links" > "unique_names"
    )
      AS "collapse_groups_by_name"
  )
    AS "collapse_groups_by_group"
)
SELECT
  "current_depth",
  "name",
  "group_id"
FROM
  "membership"
WHERE
  "current_links" = "unique_names"
ORDER BY
  "current_depth",
  "name",
  "group_id"
;

current_depth	name	group_id
4	0Z	1
4	1A	11
4	1B	11
4	1C	11
4	1D	11
4	1E	11
4	2A	21
4	2B	21
4	2C	21
4	2D	21
4	3A	31
4	3B	31
4	4A	41
4	4B	41
4	4C	41
4	4D	41
4	4E	41
4	5A	51
4	5B	51
4	5C	51
4	5D	51
4	5E	51
4	5F	51

current_depth	name	group_id
4	0Z	1
4	1A	11
4	1B	11
4	1C	11
4	1D	11
4	1E	11
4	2A	21
4	2B	21
4	2C	21
4	2D	21
4	3A	31
4	3B	31
4	4A	41
4	4B	41
4	4C	41
4	4D	41
4	4E	41
4	5A	51
4	5B	51
4	5C	51
4	5D	51
4	5E	51
4	5F	51

current_depth	name	group_id
4	0Z	1
4	1A	11
4	1B	11
4	1C	11
4	1D	11
4	1E	11
4	2A	21
4	2B	21
4	2C	21
4	2D	21
4	3A	31
4	3B	31
4	4A	41
4	4B	41
4	4C	41
4	4D	41
4	4E	41
4	5A	51
4	5B	51
4	5C	51
4	5D	51
4	5E	51
4	5F	51

current_depth	name	group_id
4	0Z	1
4	1A	11
4	1B	11
4	1C	11
4	1D	11
4	1E	11
4	2A	21
4	2B	21
4	2C	21
4	2D	21
4	3A	31
4	3B	31
4	4A	41
4	4B	41
4	4C	41
4	4D	41
4	4E	41
4	5A	51
4	5B	51
4	5C	51
4	5D	51
4	5E	51
4	5F	51