That column exists for compatibility with other databases.
Interestingly, here is information_schema.tables in PostgreSQL
postgres=# \d information_schema.tables
View "information_schema.tables"
Column | Type | Modifiers
------------------------------+-----------------------------------+-----------
table_catalog | information_schema.sql_identifier |
table_schema | information_schema.sql_identifier |
table_name | information_schema.sql_identifier |
table_type | information_schema.character_data |
self_referencing_column_name | information_schema.sql_identifier |
reference_generation | information_schema.character_data |
user_defined_type_catalog | information_schema.sql_identifier |
user_defined_type_schema | information_schema.sql_identifier |
user_defined_type_name | information_schema.sql_identifier |
is_insertable_into | information_schema.character_data |
is_typed | information_schema.character_data |
commit_action | information_schema.character_data |
View definition:
SELECT current_database()::information_schema.sql_identifier AS table_catalog, nc.nspname::information_schema.sql_identifier AS table_schema, c.relname::information_schema.sql_identifier AS table_name,
CASE
WHEN nc.oid = pg_my_temp_schema() THEN 'LOCAL TEMPORARY'::text
WHEN c.relkind = 'r'::"char" THEN 'BASE TABLE'::text
WHEN c.relkind = 'v'::"char" THEN 'VIEW'::text
ELSE NULL::text
END::information_schema.character_data AS table_type, NULL::character varying::information_schema.sql_identifier AS self_referencing_column_name, NULL::character varying::information_schema.character_data AS reference_generation, NULL::character varying::information_schema.sql_identifier AS user_defined_type_catalog, NULL::character varying::information_schema.sql_identifier AS user_defined_type_schema, NULL::character varying::information_schema.sql_identifier AS user_defined_type_name,
CASE
WHEN c.relkind = 'r'::"char" OR c.relkind = 'v'::"char" AND (EXISTS ( SELECT 1
FROM pg_rewrite
WHERE pg_rewrite.ev_class = c.oid AND pg_rewrite.ev_type = '3'::"char" AND pg_rewrite.is_instead)) THEN 'YES'::text
ELSE 'NO'::text
END::information_schema.character_data AS is_insertable_into, 'NO'::character varying::information_schema.character_data AS is_typed,
CASE
WHEN nc.oid = pg_my_temp_schema() THEN 'PRESERVE'::text
ELSE NULL::text
END::information_schema.character_data AS commit_action
FROM pg_namespace nc, pg_class c
WHERE c.relnamespace = nc.oid AND (c.relkind = ANY (ARRAY['r'::"char", 'v'::"char"])) AND NOT pg_is_other_temp_schema(nc.oid) AND (pg_has_role(c.relowner, 'USAGE'::text) OR has_table_privilege(c.oid, 'SELECT, INSERT, UPDATE, DELETE, TRUNCATE, REFERENCES, TRIGGER'::text) OR has_any_column_privilege(c.oid, 'SELECT, INSERT, UPDATE, REFERENCES'::text));
In PostgreSQL, the table_catalog field actually means something. In fact, it is part of the SQL-92 standard. The catalog of a database groups related metadata for a database. See, in PostgreSQL, the information_schema is only for the database you are connected to. In MySQL, this concept is blurred and the information_schema contains all metadata for all databases, especially if you have ALL PRIVILEGES on *.* as with root@localhost.
It makes sense that it is there in MySQL but blank. The information_schema in MySQL will collapse for you if your DB user has ALL PRIVILEGES on db.* instead of *.*. In PostgreSQL, the collapse is automatic when you connect to a database schema. So, the table_catalog column is just there for MySQL to be SQL-92 compatible and nothing more.